Research Article Extremal Unimodular Lattices in Dimension...

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Research Article Extremal Unimodular Lattices in Dimension 36 Masaaki Harada Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan Correspondence should be addressed to Masaaki Harada; [email protected] Received 15 August 2014; Accepted 6 November 2014; Published 30 November 2014 Academic Editor: Jun-Ming Xu Copyright © 2014 Masaaki Harada. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed. “Dedicated to Professor Vladimir D. Tonchev on His 60th Birthday” 1. Introduction A (Euclidean) lattice ⊂ R in dimension is unimodular if = ∗ , where the dual lattice ∗ of is defined as { ∈ R | (, ) ∈ Z for all ∈ } under the standard inner product (, ). A unimodular lattice is called even if the norm (, ) of every vector is even. A unimodular lattice, which is not even, is called odd. An even unimodular lattice in dimension exists if and only if ≡0 (mod 8), while an odd unimodular lattice exists for every dimension. Two lattices and are isomorphic, denoted by ≅ , if there exists an orthogonal matrix with =⋅, where ⋅ = { | ∈ }. e automorphism group Aut() of is the group of all orthogo- nal matrices with =⋅. Rains and Sloane [6] showed that the minimum norm min() of a unimodular lattice in dimension is bounded by min() ≤ 2⌊/24⌋ + 2 unless = 23 when min() ≤ 3. We say that a unimodular lattice meeting the upper bound is extremal. e smallest dimension for which there is an odd uni- modular lattice with minimum norm (at least) 4 is 32 (see [1]). ere are exactly five odd unimodular lattices in dimension 32 having minimum norm 4, up to isomorphism [7]. For dimensions 33, 34, and 35, the minimum norm of an odd uni- modular lattice is at most 3 (see [1]). e next dimension for which there is an odd unimodular lattice with minimum norm (at least) 4 is 36. Four extremal odd unimodular lattices in dimension 36 are known, namely, Sp4(4)D8.4 in [1], 36 in [2, Table 2], 36 in [3, Section 3], and 4 ( 36 ) in [4, Section 3]. Recently, one more lattice has been found, namely, 6 ( 36,6 ( 18 )) in [5, Table II]. is situation motivates us to improve the number of known nonisomorphic extremal odd unimodular lattices in dimension 36. e main aim of this paper is to prove the following. Proposition 1. ere are at least 26 nonisomorphic extremal odd unimodular lattices in dimension 36. e above proposition is established by constructing new extremal odd unimodular lattices in dimension 36 from self- dual Z -codes, where Z is the ring of integers modulo , by using two approaches. One approach is to consider self-dual Z 4 -codes. Let be a binary doubly even code of length 36 satisfying the following conditions: the minimum weight of is at least 16, the minimum weight of its dual code ⊥ is at least 4. (1) en a self-dual Z 4 -code with residue code gives anex- tremal odd unimodular lattice in dimension 36 by Con- struction A. We show that a binary doubly even [36, 7] code satisfying conditions (1) has weight enumerator 1 + 63 16 + 63 20 + 36 (Lemma 2). It was shown in [8] that there are four codes having the weight enumerator, up to equivalence. Hindawi Publishing Corporation International Journal of Combinatorics Volume 2014, Article ID 792471, 7 pages http://dx.doi.org/10.1155/2014/792471

Transcript of Research Article Extremal Unimodular Lattices in Dimension...

Page 1: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

Research ArticleExtremal Unimodular Lattices in Dimension 36

Masaaki Harada

Research Center for Pure and Applied Mathematics Graduate School of Information Sciences Tohoku UniversitySendai 980-8579 Japan

Correspondence should be addressed to Masaaki Harada mharadamtohokuacjp

Received 15 August 2014 Accepted 6 November 2014 Published 30 November 2014

Academic Editor Jun-Ming Xu

Copyright copy 2014 Masaaki Harada This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

New extremal odd unimodular lattices in dimension 36 are constructed Some new odd unimodular lattices in dimension 36 withlong shadows are also constructed

ldquoDedicated to Professor Vladimir D Tonchev on His 60th Birthdayrdquo

1 Introduction

A (Euclidean) lattice 119871 sub R119899 in dimension 119899 is unimodular if119871 = 119871

lowast where the dual lattice 119871lowast of 119871 is defined as 119909 isin R119899 |

(119909 119910) isin Z for all 119910 isin 119871 under the standard inner product(119909 119910) A unimodular lattice is called even if the norm (119909 119909)

of every vector 119909 is even A unimodular lattice which is noteven is called odd An even unimodular lattice in dimension119899 exists if and only if 119899 equiv 0 (mod 8) while an odd unimodularlattice exists for every dimension Two lattices 119871 and 119871

1015840 areisomorphic denoted by 119871 cong 119871

1015840 if there exists an orthogonalmatrix 119860 with 119871

1015840

= 119871 sdot 119860 where 119871 sdot 119860 = 119909119860 | 119909 isin 119871 Theautomorphism group Aut(119871) of 119871 is the group of all orthogo-nal matrices 119860 with 119871 = 119871 sdot 119860

Rains and Sloane [6] showed that the minimum normmin(119871) of a unimodular lattice 119871 in dimension 119899 is boundedby min(119871) le 2lfloor11989924rfloor + 2 unless 119899 = 23 when min(119871) le 3We say that a unimodular lattice meeting the upper bound isextremal

The smallest dimension for which there is an odd uni-modular latticewithminimumnorm (at least) 4 is 32 (see [1])There are exactly five odd unimodular lattices in dimension32 having minimum norm 4 up to isomorphism [7] Fordimensions 33 34 and 35 theminimumnormof an odd uni-modular lattice is at most 3 (see [1]) The next dimension forwhich there is an odd unimodular lattice with minimumnorm (at least) 4 is 36 Four extremal odd unimodular lattices

in dimension 36 are known namely Sp4(4)D84 in [1]11986636in

[2 Table 2] 11987336

in [3 Section 3] and 1198604(11986236) in [4

Section 3] Recently onemore lattice has been found namely1198606(119862366

(11986318)) in [5 Table II] This situation motivates us to

improve the number of known nonisomorphic extremal oddunimodular lattices in dimension 36 The main aim of thispaper is to prove the following

Proposition 1 There are at least 26 nonisomorphic extremalodd unimodular lattices in dimension 36

The above proposition is established by constructing newextremal odd unimodular lattices in dimension 36 from self-dual Z

119896-codes where Z

119896is the ring of integers modulo 119896 by

using two approaches One approach is to consider self-dualZ4-codes Let 119861 be a binary doubly even code of length 36

satisfying the following conditions

the minimum weight of 119861 is at least 16

the minimum weight of its dual code 119861perp is at least 4

(1)

Then a self-dual Z4-code with residue code 119861 gives anex-

tremal odd unimodular lattice in dimension 36 by Con-struction A We show that a binary doubly even [36 7]code satisfying conditions (1) hasweight enumerator 1 + 63119910

16

+

6311991020

+ 11991036 (Lemma 2) It was shown in [8] that there are four

codes having the weight enumerator up to equivalence

Hindawi Publishing CorporationInternational Journal of CombinatoricsVolume 2014 Article ID 792471 7 pageshttpdxdoiorg1011552014792471

2 International Journal of Combinatorics

We construct ten new extremal odd unimodular latticesin dimension 36 from self-dual Z

4-codes whose residue

codes are doubly even [36 7] codes satisfying conditions (1)(Lemma 4) New odd unimodular lattices in dimension 36

with minimum norm 3 having shadows of minimum norm 5are constructed from someof the new lattices (Proposition 7)These are often called unimodular lattices with long shadows(see [9]) The other approach is to consider self-dual Z

119896-

codes (119896 = 5 6 7 9 19) which have generator matrices of aspecial form given in (14) Eleven more new extremal oddunimodular lattices in dimension 36 are constructed by Con-struction A (Lemma 8) Finally we give a certain short obser-vation on ternary self-dual codes related to extremal oddunimodular lattices in dimension 36

All computer calculations in this paper were done byMagma [10]

2 Preliminaries

21 Unimodular Lattices Let 119871 be an odd unimodular latticeand let 119871

0denote the even sublattice that is the sublattice of

vectors of even norms Then 1198710is a sublattice of 119871 of index 2

[7] The shadow 119878(119871) of 119871 is defined to be 119871lowast

0 119871 There are

cosets 1198711 1198712 1198713of 1198710such that 119871lowast

0= 1198710cup1198711cup1198712cup1198713 where

119871 = 1198710cup 1198712and 119878 = 119871

1cup 1198713 Shadows for odd unimodular

lattices appeared in [7] and also in [11 p 440] in order toprovide restrictions on the theta series of odd unimodularlattices Two lattices 119871 and 119871

1015840 are neighbors if both latticescontain a sublattice of index 2 in common If 119871 is an odd uni-modular lattice in dimension divisible by 4 then there are twounimodular lattices containing 119871

0 which are rather 119871

namely1198710cup1198711and119871

0cup1198713Throughout this paper we denote

the two unimodular neighbors by

1198731198901(119871) = 119871

0cup 1198711 119873119890

2(119871) = 119871

0cup 1198713 (2)

The theta series 120579119871(119902) of 119871 is the formal power series

120579119871(119902) = sum

119909isin119871119902(119909119909) The kissing number of 119871 is the second

nonzero coefficient of the theta series of 119871 that is the numberof vectors of minimum norm in 119871 Conway and Sloane [7]gave some characterization of theta series of odd unimodularlattices and their shadows Using [7 (2) (3)] it is easy todetermine the possible theta series 120579

11987136(119902) and 120579

119878(11987136)(119902) of an

extremal odd unimodular lattice 11987136in dimension 36 and its

shadow 119878(11987136)

12057911987136

(119902) = 1 + (42840 + 4096120572) 1199024

+ (1916928 minus 98304120572) 1199025

+ sdot sdot sdot

(3)

120579119878(11987136)

(119902) = 120572119902 + (960 minus 60120572) 1199023

+ (3799296 + 1734120572) 1199025

+ sdot sdot sdot

(4)

respectively where 120572 is a nonnegative integer It follows fromthe coefficients of 119902 and 119902

3 in 120579119878(11987136)

(119902) that 0 le 120572 le 16

22 Self-Dual Z119896-Codes and Construction A Let Z

119896be the

ring of integersmodulo 119896 where 119896 is a positive integer greater

than 1 AZ119896-code119862 of length 119899 is aZ

119896-submodule ofZ119899

119896 Two

Z119896-codes are equivalent if one can be obtained from the other

by permuting the coordinates and (if necessary) changing thesigns of certain coordinates A code 119862 is self-dual if 119862 = 119862

perpwhere the dual code 119862

perp of 119862 is defined as 119909 isin Z119899119896| 119909 sdot 119910 = 0

for all 119910 isin 119862 under the standard inner product 119909 sdot 119910If 119862 is a self-dual Z

119896-code of length 119899 then the lattice

119860119896(119862)

=1

radic119896

(1199091 119909

119899) isin Z119899

(1199091mod 119896 119909

119899mod 119896) isin 119862

(5)

is a unimodular lattice in dimension 119899 This construction oflattices is called Construction A

3 From Self-Dual Z4-Codes

From now on we omit the term ldquooddrdquo for odd unimodularlattices in dimension 36 since all unimodular lattices indimension 36 are odd In this section we construct ten newnonisomorphic extremal unimodular lattices in dimension36 from self-dual Z

4-codes by Construction A Five new

nonisomorphic unimodular lattices in dimension 36 withminimum norm 3 having shadows of minimum norm 5 arealso constructed

31 ExtremalUnimodular Lattices EveryZ4-code119862of length

119899 has two binary codes 119862(1) and 119862(2) associated with 119862

119862(1)

= 119888 mod 2 | 119888 isin 119862

119862(2)

= 119888 mod 2 | 119888 isin Z119899

4 2119888 isin 119862

(6)

The binary codes 119862(1) and 119862(2) are called the residue and tor-

sion codes of 119862 respectively If 119862 is a self-dual Z4-code then

119862(1) is a binary doubly even code with119862

(2)

= 119862(1)perp

[12] Con-versely starting from a given binary doubly even code 119861 amethod for construction of all self-dual Z

4-codes 119862 with

119862(1)

= 119861 was given in [13 Section 3]The Euclidean weight of a codeword 119909 = (119909

1 119909

119899) of119862

is1198981(119909) + 4119898

2(119909) +119898

3(119909) where119898

120572(119909) denotes the number

of components 119894 with 119909119894

= 120572 (120572 = 1 2 3) The minimumEuclidean weight 119889

119864(119862) of 119862 is the smallest Euclidean weight

among all nonzero codewords of 119862 It is easy to see thatmin119889(119862(1)) 4119889(119862(2)) le 119889

119864(119862) where 119889(119862

(119894)

) denotes theminimum weight of 119862

(119894)

(119894 = 1 2) In addition 119889119864(119862) le

4119889(119862(2)

) and 1198604(119862) has minimum norm min4 119889

119864(119862)4

(see eg [3]) Hence if there is a binary doubly even code 119861

of length 36 satisfying conditions (1) then an extremalunimodular lattice in dimension 36 is constructed as 119860

4(119862)

through a self-dual Z4-code 119862 with 119862

(1)

= 119861 If there is abinary [36 119896] code119861 satisfying conditions (1) then 119896 = 7 or 8(see [14])

Lemma2 Let119861 be a binary doubly even [36 7] code satisfyingconditions (1) Then the weight enumerator of 119861 is 1 + 63119910

16

+

6311991020

+ 11991036

International Journal of Combinatorics 3

Proof The weight enumerator of 119861 is written as

119882119861(119910) = 1 + 119886119910

16

+ 11988711991020

+ 11988811991024

+ 11988911991028

+ 11989011991032

+ (27

minus 1 minus 119886 minus 119887 minus 119888 minus 119889 minus 119890) 11991036

(7)

where 119886 119887 119888 119889 and 119890 are nonnegative integers By theMacWilliams identity the weight enumerator of 119861perp is givenby

119882119861perp (119910) = 1 +

1

16(minus567 + 5119886 + 4119887 + 3119888 + 2119889 + 119890) 119910

+1

2(1260 minus 10119886 minus 10119887 minus 9119888 minus 7119889 minus 4119890) 119910

2

+1

16(minus112455 + 885119886 + 900119887

+ 883119888 + 770119889 + 497119890) 1199103

+ sdot sdot sdot

(8)

Since 119889(119861perp

) ge 4 the weight enumerator of 119861 is written using119886 and 119887

119882119861(119910) = 1 + 119886119910

16

+ 11988711991020

+ (882 minus 10119886 minus 4119887) 11991024

+ (minus1638 + 20119886 + 6119887) 11991028

+ (1197 minus 15119886 minus 4119887) 11991032

+ (minus314 + 4119886 + 119887) 11991036

(9)

Suppose that119861 does not contain the all-one vector 1Then119887 = 314minus4119886 In this case since the coefficients of 11991024 and 119910

28

are minus374 + 6119886 and 246 minus 4119886 these yield that 119886 ge 62 and 119886 le

61 respectively which gives the contradiction Hence 119861

contains 1 Then 119887 = 315 minus 4119886 Since the coefficient 119886 minus 63 of11991032 is 0 the weight enumerator of119861 is uniquely determined as

1 + 6311991016

+ 6311991020

+ 11991036

Remark 3 A similar approach shows that the weight enu-merator of a binary doubly even [36 8] code 119861 satisfyingconditions (1) is uniquely determined as 1+ 153119910

16

+7211991020

+

3011991024

It was shown in [8] that there are four inequivalent binary[36 7 16] codes containing 1The four codes are doubly evenHence there are exactly four binary doubly even [36 7] codessatisfying conditions (1) up to equivalenceThe four codes areoptimal in the sense that these codes achieve theGray-Rankinbound and the codewords of weight 16 are corresponding toquasi-symmetric SDP 2-(36 16 12) designs [15] Let 119861

36119894be

the binary doubly even [36 7 16] code corresponding to thequasi-symmetric SDP 2-(36 16 12) design which is theresidual design of the symmetric SDP 2-(64 28 12) design119863

119894

in [8 Section 5] (119894 = 1 2 3 4) As described above all self-dual Z

4-codes 119862 with 119862

(1)

= 11986136119894

have 119889119864(119862) = 16 (119894 =

1 2 3 4) Hence 1198604(119862) are extremal

Using the method in [13 Section 3] self-dual Z4-codes

119862 are constructed from 11986136119894

Then ten extremal unimodularlattices 119860

4(11986236119894

) (119894 = 1 2 10) are constructed where

119862(1)

36119894= 119861362

(119894 = 1 2 3) 119862(1)36119894

= 119861363

(119894 = 4 5 6 7) and119862(1)

36119894= 119861364

(119894 = 8 9 10) To distinguish between the knownlattices and our lattices we give in Table 1 the kissing numbers120591(119871) 119899

1(119871) 1198992(119871) and the orders Aut(119871) of the automor-

phism groups where 119899119894(119871) denotes the number of vectors of

norm 3 in 119873119890119894(119871) defined in (2) (119894 = 1 2) These have been

calculated by Magma Table 1 shows the following

Lemma 4 The five known lattices and the ten extremal uni-modular lattices119860

4(11986236119894

) (119894 = 1 2 10) are nonisomorphicto each other

Remark 5 In this way we have found two more extremalunimodular lattices 119860

4(119862) where 119862 are self-dual Z

4-codes

with 119862(1)

= 119861361

However we have verified by Magma thatthe two lattices are isomorphic to 119873

36in [3] and 119860

4(11986236) in

[4]

Remark 6 For 119871 = 1198604(11986236119894

) (119894 = 1 2 10) it followsfrom 120591(119871) and 119899

1(119871) 1198992(119871) that one of the two unimodular

neighbors 1198731198901(119871) and 119873119890

2(119871) defined in (2) is extremal We

have verified by Magma that the extremal one is isomorphicto 1198604(11986236119894

)

For 119894 = 1 2 10 the code 11986236119894

is equivalent to somecode 119862

36119894with generator matrix of the form

(1198687

119860 1198611+ 21198612

119874 211986822

2119863) (10)

where 119860 1198611 1198612 and 119863 are (1 0)-matrices 119868

119899denotes the

identity matrix of order 119899 and 119874 denotes the 22 times 7 zeromatrix We only list in (11) the 7 times 29 matrix 119872

119894=

(119860 1198611+ 21198612) to save space Note that (119874 2119868

222119863) in (10)

can be obtained from (1198687

119860 1198611+ 21198612) since 119862

36119894

(2)

=

11986236119894

(1)perp

A generatormatrix of1198604(11986236119894

) is obtained from thatof 11986236119894

1198721=

(((

(

01101111111000000110000333322

10101101001000111110011210223

11000011001110100111100010321

11010100101011010011012032011

00110101010101001101013312302

00000000000001111111112111333

00011111111110000001111213133

)))

)

1198722=

(((

(

01101111111000000110002111100

10101101001000111110011210023

11000011001110100111100010123

11010100101011010011012032211

00110101010101001101013312300

00000000000001111111112113131

00011111111110000001111211331

)))

)

4 International Journal of Combinatorics

Table 1 Extremal unimodular lattices in dimension 36

Lattices 119871 120591(119871) 1198991(119871) 119899

2(119871) Aut(119871)

Sp4(4)D84 in [1] 42840 480 480 3133440011986636in [2 Table 2] 42840 144 816 576

11987336in [3] 42840 0 960 849346560

1198604(11986236) in [4] 51032 0 840 660602880

1198606(119862366

(11986318)) in [5] 42840 384 576 288

1198604(119862361

) in Section 3 51032 0 840 62914561198604(119862362

) in Section 3 42840 0 960 62914561198604(119862363

) in Section 3 51032 0 840 220200961198604(119862364

) in Section 3 51032 0 840 19660801198604(119862365

) in Section 3 51032 0 840 15728641198604(119862366

) in Section 3 42840 0 960 26214401198604(119862367

) in Section 3 42840 0 960 19660801198604(119862368

) in Section 3 42840 0 960 3932161198604(119862369

) in Section 3 51032 0 840 13762561198604(1198623610

) in Section 3 51032 0 840 3932161198605(119863361

) in Section 4 42840 144 816 1441198605(119863362

) in Section 4 42840 456 504 721198606(119863363

) in Section 4 42840 240 720 2881198606(119863364

) in Section 4 42840 240 720 5761198607(119863365

) in Section 4 42840 288 672 2881198607(119863366

) in Section 4 42840 144 816 721198607(119863367

) in Section 4 42840 144 816 2881198609(119863368

) in Section 4 42840 384 576 14411986019(119863369

) in Section 4 42840 288 672 1441198605(119864361

) in Section 4 42840 456 504 721198606(119864362

) in Section 4 42840 384 576 144

1198723=

(((

(

01101111111000000110002131300

10101101001000111110011210223

11000011001110100111100010121

11010100101011010011012032011

00110101010101001101013310102

00000000000001111111112131131

00011111111110000001111233331

)))

)

1198724=

(((

(

01101111111001111101100220313

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100021110

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198725=

(((

(

01101111111001111101102202133

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100221310

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198726=

(((

(

01101111111001111101102000313

10101101001001001110101231032

11000011001110110100102101103

00111001100011100101113210223

11011000011101100011100021110

00000000000001111111112313113

00011111111110000001113103333

)))

)

1198727=

(((

(

01101111111001111101102202313

10101101001001001110101231030

11000011001110110100102101121

00111001100011100101113210201

11011000011101100011100021112

00000000000001111111112111113

00011111111110000001113301333

)))

)

1198728=

(((

(

11111110000111111111000022313

11110001100111000000001130313

01000110111001110100011231021

10011100001011100001101233011

01010101010010101101003301321

00000000011111111111112331111

01111111100000000111111233313

)))

)

International Journal of Combinatorics 5

1198729=

(((

(

11111110000111111111002200111

11110001100111000000001332311

01000110111001110100013211021

10011100001011100001103213011

01010101010010101101000331321

01111111100000000111111211113

00000000011111111111111303313

)))

)

11987210

=

(((

(

11111110000111111111002202313

11110001100111000000001130311

01000110111001110100011211021

10011100001011100001101213011

01010101010010101101003321123

00000000011111111111112313313

01111111100000000111111211111

)))

)

(11)

32 Unimodular Lattices with Long Shadows The possibletheta series of a unimodular lattice 119871 in dimension 36 havingminimum norm 3 and its shadow are as follows

1 + (960 minus 120572) 1199023

+ (42840 + 4096120573) 1199024

+ sdot sdot sdot

120573119902 + (120572 minus 60120573) 1199023

+ (3833856 minus 36120572 + 1734120573) 1199025

+ sdot sdot sdot

(12)

respectively where 120572 and 120573 are integers with 0 le 120573 le 12057260 lt

16 [3] Then the kissing number of 119871 is at most 960 andmin(119878(119871)) le 5 Unimodular lattices 119871 with min(119871) = 3 andmin(119878(119871)) = 5 are often called unimodular lattices with longshadows (see [9]) Only one unimodular lattice 119871 in dimen-sion 36 with min(119871) = 3 and min(119878(119871)) = 5 was knownnamely 119860

4(11986236) in [3]

Let 119871 be one of 1198604(119862362

) 1198604(119862366

) 1198604(119862367

) and1198604(119862368

) Since 1198991(119871) 1198992(119871) = 0 960 one of the two

unimodular neighbors 1198731198901(119871) and 119873119890

2(119871) in (2) is extremal

and the other is a unimodular lattice 1198711015840 with minimum norm

3 having shadow of minimum norm 5 We denote such lat-tices 1198711015840 constructed from119860

4(119862362

)1198604(119862366

)1198604(119862367

) and1198604(119862368

) by119873361

119873362

119873363

and119873364

respectivelyWe listin Table 2 the orders Aut(119873

36119894) (119894 = 1 2 3 4) of the auto-

morphism groups which have been calculated by MagmaTable 2 shows the following

Proposition7 There are at least 5 nonisomorphic unimodularlattices 119871 in dimension 36 withmin(119871) = 3 andmin(119878(119871)) = 5

4 From Self-Dual Z119896-Codes (119896 ge 5)

In this section we construct more extremal unimodularlattices in dimension 36 from self-dual Z

119896-codes (119896 ge 5)

Let119860119879 denote the transpose of amatrix119860 An 119899times119899matrixis negacirculant if it has the following form

(

1199030

1199031

sdot sdot sdot 119903119899minus1

minus119903119899minus1

1199030

sdot sdot sdot 119903119899minus2

minus119903119899minus2

minus119903119899minus1

sdot sdot sdot 119903119899minus3

minus1199031

minus1199032

sdot sdot sdot 1199030

) (13)

Table 2 Aut(11987336119894

) (119894 = 1 2 3 4)

Lattices 119871 Aut(119871)1198604(11986236) in [3] 1698693120

119873361

12582912119873362

5242880119873363

3932160119873364

786432

Let 11986336119894

(119894 = 1 2 9) and 11986436119894

(119894 = 1 2) be Z119896-codes of

length 36 with generator matrices of the following form

(11986818

119860 119861

minus119861119879

119860119879 ) (14)

where 119896 are listed in Table 3 and 119860 and 119861 are 9 times 9

negacirculant matrices with first rows 119903119860and 119903

119861listed in

Table 3 It is easy to see that these codes are self-dual since119860119860119879

+ 119861119861119879

= minus1198689 Thus 119860

119896(11986336119894

) (119894 = 1 2 9) and119860119896(11986436119894

) (119894 = 1 2) are unimodular lattices for 119896 given inTable 3 In addition we have verified by Magma that theselattices are extremal

To distinguish between the above eleven lattices and the15 known lattices in Table 1 we give 120591(119871) 119899

1(119871) 1198992(119871)

and Aut(119871) which have been calculated by MagmaThe two lattices have the identical (120591(119871) 119899

1(119871) 1198992(119871)

Aut(119871)) for each of the pairs (1198605(119864361

) 1198605(119863362

)) and(1198606(119864362

) 1198609(119863368

)) However we have verified by Magmathat the two lattices are nonisomorphic for each pair There-fore we have the following

Lemma 8 The 26 lattices in Table 1 are nonisomorphic to eachother

Lemma 8 establishes Proposition 1

Remark 9 Similar to Remark 6 it is known [3] that theextremal neighbor is isomorphic to 119871 for the case where 119871 is11987336in [3] and we have verified by Magma that the extremal

neighbor is isomorphic to 119871 for the case where 119871 is 1198604(11986236)

in [4]

5 Related Ternary Self-Dual Codes

In this section we give a certain short observation on ternaryself-dual codes related to some extremal odd unimodularlattices in dimension 36

51 Unimodular Lattices from Ternary Self-Dual Codes Let11987936

be a ternary self-dual code of length 36 The two uni-modular neighbors 119873119890

1(1198603(11987936)) and 119873119890

2(1198603(11987936)) given

in (2) are described in [16] as 119871119878(11987936) and 119871

119879(11987936) In this

section we use the notation 119871119878(11987936) and 119871

119879(11987936) instead of

1198731198901(1198603(11987936)) and 119873119890

2(1198603(11987936)) since the explicit construc-

tions and some properties of 119871119878(11987936) and 119871

119879(11987936) are given in

[16] By Theorem 6 in [16] (see also Theorem 31 in [2])119871119879(11987936) is extremal when119879

36satisfies the following condition

6 International Journal of Combinatorics

Table 3 Self-dual Z119896-codes of length 36

Codes 119896 119903119860

119903119861

119863361

5 (0 0 0 1 2 2 0 4 2) (0 0 0 0 4 3 3 0 1)

119863362

5 (0 0 0 1 3 0 2 0 4) (3 0 4 1 4 0 0 4 4)

119863363

6 (0 1 5 3 2 0 3 5 1) (3 1 0 0 5 1 1 1 1)

119863364

6 (0 1 3 5 1 5 5 4 4) (4 0 3 2 4 5 5 2 4)

119863365

7 (0 1 6 3 5 0 4 5 4) (0 1 6 3 5 2 1 5 1)

119863366

7 (0 1 1 3 2 6 1 4 6) (0 1 4 0 5 3 6 2 0)

119863367

7 (0 0 0 1 5 5 5 1 1) (0 5 4 2 5 1 1 3 6)

119863368

9 (0 0 0 1 0 4 3 0 0) (0 4 1 5 3 5 1 7 0)

119863369

19 (0 0 0 1 15 15 9 6 5) (14 16 0 14 15 8 8 3 12)

119864361

5 (0 1 0 2 1 3 2 2 0) (2 0 1 0 1 1 2 3 1)

119864362

6 (0 1 5 3 4 4 1 1 0) (4 0 1 3 4 2 3 0 1)

(a) and both 119871119878(11987936) and 119871

119879(11987936) are extremal when 119879

36

satisfies the following condition (b)(a) extremal (minimum weight 12) and admissible (the

number of 1rsquos in the components of every codewordof weight 36 is even)

(b) minimum weight 9 and maximum weight 33For each of (a) and (b) no ternary self-dual code satisfyingthe condition is currently known

52 Condition (a) Suppose that 11987936

satisfies condition (a)Since 119879

36has minimum weight 12 119860

3(11987936) has minimum

norm 3 and kissing number 72 By Theorem 6 in [16]min(119871

119879(11987936)) = 4 and min(119871

119878(11987936)) = 3 Hence since the

shadow of 119871119879(11987936) contains no vector of norm 1 by (3) and

(4) 119871119879(11987936) has theta series 1+ 42840119902

4

+ 19169281199025

+ sdot sdot sdot Itfollows that 119899

1(119871119879(11987936)) 1198992(119871119879(11987936)) = 72 888

By Theorem 1 in [17] the possible complete weightenumerator 119882

119862(119909 119910 119911) of a ternary extremal self-dual code

119862 of length 36 containing 1 is written as

119886112057536

+ 11988621205723

12+ 11988631205722

121205732

6+ 119886412057212

(1205732

6)2

+ 1198865(1205732

6)3

+ 119886612057361205741812057212

+ 11988671205736120574181205732

6

(15)

using some 119886119894isin R (119894 = 1 2 7) where 120572

12= 119886(119886

3

+ 81199013

)1205736= 1198862

minus12119887 12057418

= 1198866

minus201198863

1199013

minus81199016 and 120575

36= 1199013

(1198863

minus1199013

)3

and 119886 = 1199093

+ 1199103

+ 1199113 119901 = 3119909119910119911 and 119887 = 119909

3

1199103

+ 1199093

1199113

+ 1199103

1199113

From the minimum weight we have the following

1198862=

3281

13824minus

1198861

64 119886

3=

203

4608minus

91198861

256

1198864=

1763

13824+

31198861

128 119886

5= minus

277

13824minus

1198861

256

1198866=

1133

1728+

31198861

64 119886

7= minus

77

1728minus

1198861

64

(16)

Since 119882119862(119909 119910 119911) contains the term (15180 + 2916119886

1)11991015

11991121

if 119862 is admissible then

1198861= minus

15180

2916 (17)

Hence the complete weight enumerator of a ternary admis-sible extremal self-dual code containing 1 is uniquely deter-mined which is listed in (18)

11990936

+ 11991036

+ 11991136

+ 7870626011990912

11991012

11991112

+ 682 (11990918

11991018

+ 11990918

11991118

+ 11991018

11991118

)

+ 7019232 (11990915

11991015

1199116

+ 11990915

1199106

11991115

+ 1199096

11991015

11991115

)

+ 29172 (11990924

1199106

1199116

+ 1199096

11991024

1199116

+ 1199096

1199106

11991124

)

+ 10260316 (11990918

1199109

1199119

+ 1199099

11991018

1199119

+ 1199099

1199109

11991118

)

+ 37995408 (11990912

11991015

1199119

+ 11990912

1199109

11991115

+ 11990915

11991012

1199119

+11990915

1199109

11991112

+ 1199099

11991012

11991115

+ 1199099

11991015

11991112

)

+ 3924756 (11990912

11991018

1199116

+ 11990912

1199106

11991118

+ 11990918

11991012

1199116

+11990918

1199106

11991112

+ 1199096

11991012

11991118

+ 1199096

11991018

11991112

)

+ 58344 (11990912

11991021

1199113

+ 11990912

1199103

11991121

+ 11990921

11991012

1199113

+11990921

1199103

11991112

+ 1199093

11991012

11991121

+ 1199093

11991021

11991112

)

+ 102 (11990912

11991024

+ 11990912

11991124

+ 11990924

11991012

+11990924

11991112

+ 11991012

11991124

+ 11991024

11991112

)

+ 170544 (11990915

11991018

1199113

+ 11990915

1199103

11991118

+ 11990918

11991015

1199113

+11990918

1199103

11991115

+ 1199093

11991015

11991118

+ 1199093

11991018

11991115

)

+ 641784 (11990921

1199106

1199119

+ 11990921

1199109

1199116

+ 1199096

11991021

1199119

+1199096

1199109

11991121

+ 1199099

11991021

1199116

+ 1199099

1199106

11991121

)

+ 6732 (11990924

1199103

1199119

+ 11990924

1199109

1199113

+ 1199093

11991024

1199119

+1199093

1199109

11991124

+ 1199099

11991024

1199113

+ 1199099

1199103

11991124

)

(18)

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

2 International Journal of Combinatorics

We construct ten new extremal odd unimodular latticesin dimension 36 from self-dual Z

4-codes whose residue

codes are doubly even [36 7] codes satisfying conditions (1)(Lemma 4) New odd unimodular lattices in dimension 36

with minimum norm 3 having shadows of minimum norm 5are constructed from someof the new lattices (Proposition 7)These are often called unimodular lattices with long shadows(see [9]) The other approach is to consider self-dual Z

119896-

codes (119896 = 5 6 7 9 19) which have generator matrices of aspecial form given in (14) Eleven more new extremal oddunimodular lattices in dimension 36 are constructed by Con-struction A (Lemma 8) Finally we give a certain short obser-vation on ternary self-dual codes related to extremal oddunimodular lattices in dimension 36

All computer calculations in this paper were done byMagma [10]

2 Preliminaries

21 Unimodular Lattices Let 119871 be an odd unimodular latticeand let 119871

0denote the even sublattice that is the sublattice of

vectors of even norms Then 1198710is a sublattice of 119871 of index 2

[7] The shadow 119878(119871) of 119871 is defined to be 119871lowast

0 119871 There are

cosets 1198711 1198712 1198713of 1198710such that 119871lowast

0= 1198710cup1198711cup1198712cup1198713 where

119871 = 1198710cup 1198712and 119878 = 119871

1cup 1198713 Shadows for odd unimodular

lattices appeared in [7] and also in [11 p 440] in order toprovide restrictions on the theta series of odd unimodularlattices Two lattices 119871 and 119871

1015840 are neighbors if both latticescontain a sublattice of index 2 in common If 119871 is an odd uni-modular lattice in dimension divisible by 4 then there are twounimodular lattices containing 119871

0 which are rather 119871

namely1198710cup1198711and119871

0cup1198713Throughout this paper we denote

the two unimodular neighbors by

1198731198901(119871) = 119871

0cup 1198711 119873119890

2(119871) = 119871

0cup 1198713 (2)

The theta series 120579119871(119902) of 119871 is the formal power series

120579119871(119902) = sum

119909isin119871119902(119909119909) The kissing number of 119871 is the second

nonzero coefficient of the theta series of 119871 that is the numberof vectors of minimum norm in 119871 Conway and Sloane [7]gave some characterization of theta series of odd unimodularlattices and their shadows Using [7 (2) (3)] it is easy todetermine the possible theta series 120579

11987136(119902) and 120579

119878(11987136)(119902) of an

extremal odd unimodular lattice 11987136in dimension 36 and its

shadow 119878(11987136)

12057911987136

(119902) = 1 + (42840 + 4096120572) 1199024

+ (1916928 minus 98304120572) 1199025

+ sdot sdot sdot

(3)

120579119878(11987136)

(119902) = 120572119902 + (960 minus 60120572) 1199023

+ (3799296 + 1734120572) 1199025

+ sdot sdot sdot

(4)

respectively where 120572 is a nonnegative integer It follows fromthe coefficients of 119902 and 119902

3 in 120579119878(11987136)

(119902) that 0 le 120572 le 16

22 Self-Dual Z119896-Codes and Construction A Let Z

119896be the

ring of integersmodulo 119896 where 119896 is a positive integer greater

than 1 AZ119896-code119862 of length 119899 is aZ

119896-submodule ofZ119899

119896 Two

Z119896-codes are equivalent if one can be obtained from the other

by permuting the coordinates and (if necessary) changing thesigns of certain coordinates A code 119862 is self-dual if 119862 = 119862

perpwhere the dual code 119862

perp of 119862 is defined as 119909 isin Z119899119896| 119909 sdot 119910 = 0

for all 119910 isin 119862 under the standard inner product 119909 sdot 119910If 119862 is a self-dual Z

119896-code of length 119899 then the lattice

119860119896(119862)

=1

radic119896

(1199091 119909

119899) isin Z119899

(1199091mod 119896 119909

119899mod 119896) isin 119862

(5)

is a unimodular lattice in dimension 119899 This construction oflattices is called Construction A

3 From Self-Dual Z4-Codes

From now on we omit the term ldquooddrdquo for odd unimodularlattices in dimension 36 since all unimodular lattices indimension 36 are odd In this section we construct ten newnonisomorphic extremal unimodular lattices in dimension36 from self-dual Z

4-codes by Construction A Five new

nonisomorphic unimodular lattices in dimension 36 withminimum norm 3 having shadows of minimum norm 5 arealso constructed

31 ExtremalUnimodular Lattices EveryZ4-code119862of length

119899 has two binary codes 119862(1) and 119862(2) associated with 119862

119862(1)

= 119888 mod 2 | 119888 isin 119862

119862(2)

= 119888 mod 2 | 119888 isin Z119899

4 2119888 isin 119862

(6)

The binary codes 119862(1) and 119862(2) are called the residue and tor-

sion codes of 119862 respectively If 119862 is a self-dual Z4-code then

119862(1) is a binary doubly even code with119862

(2)

= 119862(1)perp

[12] Con-versely starting from a given binary doubly even code 119861 amethod for construction of all self-dual Z

4-codes 119862 with

119862(1)

= 119861 was given in [13 Section 3]The Euclidean weight of a codeword 119909 = (119909

1 119909

119899) of119862

is1198981(119909) + 4119898

2(119909) +119898

3(119909) where119898

120572(119909) denotes the number

of components 119894 with 119909119894

= 120572 (120572 = 1 2 3) The minimumEuclidean weight 119889

119864(119862) of 119862 is the smallest Euclidean weight

among all nonzero codewords of 119862 It is easy to see thatmin119889(119862(1)) 4119889(119862(2)) le 119889

119864(119862) where 119889(119862

(119894)

) denotes theminimum weight of 119862

(119894)

(119894 = 1 2) In addition 119889119864(119862) le

4119889(119862(2)

) and 1198604(119862) has minimum norm min4 119889

119864(119862)4

(see eg [3]) Hence if there is a binary doubly even code 119861

of length 36 satisfying conditions (1) then an extremalunimodular lattice in dimension 36 is constructed as 119860

4(119862)

through a self-dual Z4-code 119862 with 119862

(1)

= 119861 If there is abinary [36 119896] code119861 satisfying conditions (1) then 119896 = 7 or 8(see [14])

Lemma2 Let119861 be a binary doubly even [36 7] code satisfyingconditions (1) Then the weight enumerator of 119861 is 1 + 63119910

16

+

6311991020

+ 11991036

International Journal of Combinatorics 3

Proof The weight enumerator of 119861 is written as

119882119861(119910) = 1 + 119886119910

16

+ 11988711991020

+ 11988811991024

+ 11988911991028

+ 11989011991032

+ (27

minus 1 minus 119886 minus 119887 minus 119888 minus 119889 minus 119890) 11991036

(7)

where 119886 119887 119888 119889 and 119890 are nonnegative integers By theMacWilliams identity the weight enumerator of 119861perp is givenby

119882119861perp (119910) = 1 +

1

16(minus567 + 5119886 + 4119887 + 3119888 + 2119889 + 119890) 119910

+1

2(1260 minus 10119886 minus 10119887 minus 9119888 minus 7119889 minus 4119890) 119910

2

+1

16(minus112455 + 885119886 + 900119887

+ 883119888 + 770119889 + 497119890) 1199103

+ sdot sdot sdot

(8)

Since 119889(119861perp

) ge 4 the weight enumerator of 119861 is written using119886 and 119887

119882119861(119910) = 1 + 119886119910

16

+ 11988711991020

+ (882 minus 10119886 minus 4119887) 11991024

+ (minus1638 + 20119886 + 6119887) 11991028

+ (1197 minus 15119886 minus 4119887) 11991032

+ (minus314 + 4119886 + 119887) 11991036

(9)

Suppose that119861 does not contain the all-one vector 1Then119887 = 314minus4119886 In this case since the coefficients of 11991024 and 119910

28

are minus374 + 6119886 and 246 minus 4119886 these yield that 119886 ge 62 and 119886 le

61 respectively which gives the contradiction Hence 119861

contains 1 Then 119887 = 315 minus 4119886 Since the coefficient 119886 minus 63 of11991032 is 0 the weight enumerator of119861 is uniquely determined as

1 + 6311991016

+ 6311991020

+ 11991036

Remark 3 A similar approach shows that the weight enu-merator of a binary doubly even [36 8] code 119861 satisfyingconditions (1) is uniquely determined as 1+ 153119910

16

+7211991020

+

3011991024

It was shown in [8] that there are four inequivalent binary[36 7 16] codes containing 1The four codes are doubly evenHence there are exactly four binary doubly even [36 7] codessatisfying conditions (1) up to equivalenceThe four codes areoptimal in the sense that these codes achieve theGray-Rankinbound and the codewords of weight 16 are corresponding toquasi-symmetric SDP 2-(36 16 12) designs [15] Let 119861

36119894be

the binary doubly even [36 7 16] code corresponding to thequasi-symmetric SDP 2-(36 16 12) design which is theresidual design of the symmetric SDP 2-(64 28 12) design119863

119894

in [8 Section 5] (119894 = 1 2 3 4) As described above all self-dual Z

4-codes 119862 with 119862

(1)

= 11986136119894

have 119889119864(119862) = 16 (119894 =

1 2 3 4) Hence 1198604(119862) are extremal

Using the method in [13 Section 3] self-dual Z4-codes

119862 are constructed from 11986136119894

Then ten extremal unimodularlattices 119860

4(11986236119894

) (119894 = 1 2 10) are constructed where

119862(1)

36119894= 119861362

(119894 = 1 2 3) 119862(1)36119894

= 119861363

(119894 = 4 5 6 7) and119862(1)

36119894= 119861364

(119894 = 8 9 10) To distinguish between the knownlattices and our lattices we give in Table 1 the kissing numbers120591(119871) 119899

1(119871) 1198992(119871) and the orders Aut(119871) of the automor-

phism groups where 119899119894(119871) denotes the number of vectors of

norm 3 in 119873119890119894(119871) defined in (2) (119894 = 1 2) These have been

calculated by Magma Table 1 shows the following

Lemma 4 The five known lattices and the ten extremal uni-modular lattices119860

4(11986236119894

) (119894 = 1 2 10) are nonisomorphicto each other

Remark 5 In this way we have found two more extremalunimodular lattices 119860

4(119862) where 119862 are self-dual Z

4-codes

with 119862(1)

= 119861361

However we have verified by Magma thatthe two lattices are isomorphic to 119873

36in [3] and 119860

4(11986236) in

[4]

Remark 6 For 119871 = 1198604(11986236119894

) (119894 = 1 2 10) it followsfrom 120591(119871) and 119899

1(119871) 1198992(119871) that one of the two unimodular

neighbors 1198731198901(119871) and 119873119890

2(119871) defined in (2) is extremal We

have verified by Magma that the extremal one is isomorphicto 1198604(11986236119894

)

For 119894 = 1 2 10 the code 11986236119894

is equivalent to somecode 119862

36119894with generator matrix of the form

(1198687

119860 1198611+ 21198612

119874 211986822

2119863) (10)

where 119860 1198611 1198612 and 119863 are (1 0)-matrices 119868

119899denotes the

identity matrix of order 119899 and 119874 denotes the 22 times 7 zeromatrix We only list in (11) the 7 times 29 matrix 119872

119894=

(119860 1198611+ 21198612) to save space Note that (119874 2119868

222119863) in (10)

can be obtained from (1198687

119860 1198611+ 21198612) since 119862

36119894

(2)

=

11986236119894

(1)perp

A generatormatrix of1198604(11986236119894

) is obtained from thatof 11986236119894

1198721=

(((

(

01101111111000000110000333322

10101101001000111110011210223

11000011001110100111100010321

11010100101011010011012032011

00110101010101001101013312302

00000000000001111111112111333

00011111111110000001111213133

)))

)

1198722=

(((

(

01101111111000000110002111100

10101101001000111110011210023

11000011001110100111100010123

11010100101011010011012032211

00110101010101001101013312300

00000000000001111111112113131

00011111111110000001111211331

)))

)

4 International Journal of Combinatorics

Table 1 Extremal unimodular lattices in dimension 36

Lattices 119871 120591(119871) 1198991(119871) 119899

2(119871) Aut(119871)

Sp4(4)D84 in [1] 42840 480 480 3133440011986636in [2 Table 2] 42840 144 816 576

11987336in [3] 42840 0 960 849346560

1198604(11986236) in [4] 51032 0 840 660602880

1198606(119862366

(11986318)) in [5] 42840 384 576 288

1198604(119862361

) in Section 3 51032 0 840 62914561198604(119862362

) in Section 3 42840 0 960 62914561198604(119862363

) in Section 3 51032 0 840 220200961198604(119862364

) in Section 3 51032 0 840 19660801198604(119862365

) in Section 3 51032 0 840 15728641198604(119862366

) in Section 3 42840 0 960 26214401198604(119862367

) in Section 3 42840 0 960 19660801198604(119862368

) in Section 3 42840 0 960 3932161198604(119862369

) in Section 3 51032 0 840 13762561198604(1198623610

) in Section 3 51032 0 840 3932161198605(119863361

) in Section 4 42840 144 816 1441198605(119863362

) in Section 4 42840 456 504 721198606(119863363

) in Section 4 42840 240 720 2881198606(119863364

) in Section 4 42840 240 720 5761198607(119863365

) in Section 4 42840 288 672 2881198607(119863366

) in Section 4 42840 144 816 721198607(119863367

) in Section 4 42840 144 816 2881198609(119863368

) in Section 4 42840 384 576 14411986019(119863369

) in Section 4 42840 288 672 1441198605(119864361

) in Section 4 42840 456 504 721198606(119864362

) in Section 4 42840 384 576 144

1198723=

(((

(

01101111111000000110002131300

10101101001000111110011210223

11000011001110100111100010121

11010100101011010011012032011

00110101010101001101013310102

00000000000001111111112131131

00011111111110000001111233331

)))

)

1198724=

(((

(

01101111111001111101100220313

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100021110

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198725=

(((

(

01101111111001111101102202133

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100221310

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198726=

(((

(

01101111111001111101102000313

10101101001001001110101231032

11000011001110110100102101103

00111001100011100101113210223

11011000011101100011100021110

00000000000001111111112313113

00011111111110000001113103333

)))

)

1198727=

(((

(

01101111111001111101102202313

10101101001001001110101231030

11000011001110110100102101121

00111001100011100101113210201

11011000011101100011100021112

00000000000001111111112111113

00011111111110000001113301333

)))

)

1198728=

(((

(

11111110000111111111000022313

11110001100111000000001130313

01000110111001110100011231021

10011100001011100001101233011

01010101010010101101003301321

00000000011111111111112331111

01111111100000000111111233313

)))

)

International Journal of Combinatorics 5

1198729=

(((

(

11111110000111111111002200111

11110001100111000000001332311

01000110111001110100013211021

10011100001011100001103213011

01010101010010101101000331321

01111111100000000111111211113

00000000011111111111111303313

)))

)

11987210

=

(((

(

11111110000111111111002202313

11110001100111000000001130311

01000110111001110100011211021

10011100001011100001101213011

01010101010010101101003321123

00000000011111111111112313313

01111111100000000111111211111

)))

)

(11)

32 Unimodular Lattices with Long Shadows The possibletheta series of a unimodular lattice 119871 in dimension 36 havingminimum norm 3 and its shadow are as follows

1 + (960 minus 120572) 1199023

+ (42840 + 4096120573) 1199024

+ sdot sdot sdot

120573119902 + (120572 minus 60120573) 1199023

+ (3833856 minus 36120572 + 1734120573) 1199025

+ sdot sdot sdot

(12)

respectively where 120572 and 120573 are integers with 0 le 120573 le 12057260 lt

16 [3] Then the kissing number of 119871 is at most 960 andmin(119878(119871)) le 5 Unimodular lattices 119871 with min(119871) = 3 andmin(119878(119871)) = 5 are often called unimodular lattices with longshadows (see [9]) Only one unimodular lattice 119871 in dimen-sion 36 with min(119871) = 3 and min(119878(119871)) = 5 was knownnamely 119860

4(11986236) in [3]

Let 119871 be one of 1198604(119862362

) 1198604(119862366

) 1198604(119862367

) and1198604(119862368

) Since 1198991(119871) 1198992(119871) = 0 960 one of the two

unimodular neighbors 1198731198901(119871) and 119873119890

2(119871) in (2) is extremal

and the other is a unimodular lattice 1198711015840 with minimum norm

3 having shadow of minimum norm 5 We denote such lat-tices 1198711015840 constructed from119860

4(119862362

)1198604(119862366

)1198604(119862367

) and1198604(119862368

) by119873361

119873362

119873363

and119873364

respectivelyWe listin Table 2 the orders Aut(119873

36119894) (119894 = 1 2 3 4) of the auto-

morphism groups which have been calculated by MagmaTable 2 shows the following

Proposition7 There are at least 5 nonisomorphic unimodularlattices 119871 in dimension 36 withmin(119871) = 3 andmin(119878(119871)) = 5

4 From Self-Dual Z119896-Codes (119896 ge 5)

In this section we construct more extremal unimodularlattices in dimension 36 from self-dual Z

119896-codes (119896 ge 5)

Let119860119879 denote the transpose of amatrix119860 An 119899times119899matrixis negacirculant if it has the following form

(

1199030

1199031

sdot sdot sdot 119903119899minus1

minus119903119899minus1

1199030

sdot sdot sdot 119903119899minus2

minus119903119899minus2

minus119903119899minus1

sdot sdot sdot 119903119899minus3

minus1199031

minus1199032

sdot sdot sdot 1199030

) (13)

Table 2 Aut(11987336119894

) (119894 = 1 2 3 4)

Lattices 119871 Aut(119871)1198604(11986236) in [3] 1698693120

119873361

12582912119873362

5242880119873363

3932160119873364

786432

Let 11986336119894

(119894 = 1 2 9) and 11986436119894

(119894 = 1 2) be Z119896-codes of

length 36 with generator matrices of the following form

(11986818

119860 119861

minus119861119879

119860119879 ) (14)

where 119896 are listed in Table 3 and 119860 and 119861 are 9 times 9

negacirculant matrices with first rows 119903119860and 119903

119861listed in

Table 3 It is easy to see that these codes are self-dual since119860119860119879

+ 119861119861119879

= minus1198689 Thus 119860

119896(11986336119894

) (119894 = 1 2 9) and119860119896(11986436119894

) (119894 = 1 2) are unimodular lattices for 119896 given inTable 3 In addition we have verified by Magma that theselattices are extremal

To distinguish between the above eleven lattices and the15 known lattices in Table 1 we give 120591(119871) 119899

1(119871) 1198992(119871)

and Aut(119871) which have been calculated by MagmaThe two lattices have the identical (120591(119871) 119899

1(119871) 1198992(119871)

Aut(119871)) for each of the pairs (1198605(119864361

) 1198605(119863362

)) and(1198606(119864362

) 1198609(119863368

)) However we have verified by Magmathat the two lattices are nonisomorphic for each pair There-fore we have the following

Lemma 8 The 26 lattices in Table 1 are nonisomorphic to eachother

Lemma 8 establishes Proposition 1

Remark 9 Similar to Remark 6 it is known [3] that theextremal neighbor is isomorphic to 119871 for the case where 119871 is11987336in [3] and we have verified by Magma that the extremal

neighbor is isomorphic to 119871 for the case where 119871 is 1198604(11986236)

in [4]

5 Related Ternary Self-Dual Codes

In this section we give a certain short observation on ternaryself-dual codes related to some extremal odd unimodularlattices in dimension 36

51 Unimodular Lattices from Ternary Self-Dual Codes Let11987936

be a ternary self-dual code of length 36 The two uni-modular neighbors 119873119890

1(1198603(11987936)) and 119873119890

2(1198603(11987936)) given

in (2) are described in [16] as 119871119878(11987936) and 119871

119879(11987936) In this

section we use the notation 119871119878(11987936) and 119871

119879(11987936) instead of

1198731198901(1198603(11987936)) and 119873119890

2(1198603(11987936)) since the explicit construc-

tions and some properties of 119871119878(11987936) and 119871

119879(11987936) are given in

[16] By Theorem 6 in [16] (see also Theorem 31 in [2])119871119879(11987936) is extremal when119879

36satisfies the following condition

6 International Journal of Combinatorics

Table 3 Self-dual Z119896-codes of length 36

Codes 119896 119903119860

119903119861

119863361

5 (0 0 0 1 2 2 0 4 2) (0 0 0 0 4 3 3 0 1)

119863362

5 (0 0 0 1 3 0 2 0 4) (3 0 4 1 4 0 0 4 4)

119863363

6 (0 1 5 3 2 0 3 5 1) (3 1 0 0 5 1 1 1 1)

119863364

6 (0 1 3 5 1 5 5 4 4) (4 0 3 2 4 5 5 2 4)

119863365

7 (0 1 6 3 5 0 4 5 4) (0 1 6 3 5 2 1 5 1)

119863366

7 (0 1 1 3 2 6 1 4 6) (0 1 4 0 5 3 6 2 0)

119863367

7 (0 0 0 1 5 5 5 1 1) (0 5 4 2 5 1 1 3 6)

119863368

9 (0 0 0 1 0 4 3 0 0) (0 4 1 5 3 5 1 7 0)

119863369

19 (0 0 0 1 15 15 9 6 5) (14 16 0 14 15 8 8 3 12)

119864361

5 (0 1 0 2 1 3 2 2 0) (2 0 1 0 1 1 2 3 1)

119864362

6 (0 1 5 3 4 4 1 1 0) (4 0 1 3 4 2 3 0 1)

(a) and both 119871119878(11987936) and 119871

119879(11987936) are extremal when 119879

36

satisfies the following condition (b)(a) extremal (minimum weight 12) and admissible (the

number of 1rsquos in the components of every codewordof weight 36 is even)

(b) minimum weight 9 and maximum weight 33For each of (a) and (b) no ternary self-dual code satisfyingthe condition is currently known

52 Condition (a) Suppose that 11987936

satisfies condition (a)Since 119879

36has minimum weight 12 119860

3(11987936) has minimum

norm 3 and kissing number 72 By Theorem 6 in [16]min(119871

119879(11987936)) = 4 and min(119871

119878(11987936)) = 3 Hence since the

shadow of 119871119879(11987936) contains no vector of norm 1 by (3) and

(4) 119871119879(11987936) has theta series 1+ 42840119902

4

+ 19169281199025

+ sdot sdot sdot Itfollows that 119899

1(119871119879(11987936)) 1198992(119871119879(11987936)) = 72 888

By Theorem 1 in [17] the possible complete weightenumerator 119882

119862(119909 119910 119911) of a ternary extremal self-dual code

119862 of length 36 containing 1 is written as

119886112057536

+ 11988621205723

12+ 11988631205722

121205732

6+ 119886412057212

(1205732

6)2

+ 1198865(1205732

6)3

+ 119886612057361205741812057212

+ 11988671205736120574181205732

6

(15)

using some 119886119894isin R (119894 = 1 2 7) where 120572

12= 119886(119886

3

+ 81199013

)1205736= 1198862

minus12119887 12057418

= 1198866

minus201198863

1199013

minus81199016 and 120575

36= 1199013

(1198863

minus1199013

)3

and 119886 = 1199093

+ 1199103

+ 1199113 119901 = 3119909119910119911 and 119887 = 119909

3

1199103

+ 1199093

1199113

+ 1199103

1199113

From the minimum weight we have the following

1198862=

3281

13824minus

1198861

64 119886

3=

203

4608minus

91198861

256

1198864=

1763

13824+

31198861

128 119886

5= minus

277

13824minus

1198861

256

1198866=

1133

1728+

31198861

64 119886

7= minus

77

1728minus

1198861

64

(16)

Since 119882119862(119909 119910 119911) contains the term (15180 + 2916119886

1)11991015

11991121

if 119862 is admissible then

1198861= minus

15180

2916 (17)

Hence the complete weight enumerator of a ternary admis-sible extremal self-dual code containing 1 is uniquely deter-mined which is listed in (18)

11990936

+ 11991036

+ 11991136

+ 7870626011990912

11991012

11991112

+ 682 (11990918

11991018

+ 11990918

11991118

+ 11991018

11991118

)

+ 7019232 (11990915

11991015

1199116

+ 11990915

1199106

11991115

+ 1199096

11991015

11991115

)

+ 29172 (11990924

1199106

1199116

+ 1199096

11991024

1199116

+ 1199096

1199106

11991124

)

+ 10260316 (11990918

1199109

1199119

+ 1199099

11991018

1199119

+ 1199099

1199109

11991118

)

+ 37995408 (11990912

11991015

1199119

+ 11990912

1199109

11991115

+ 11990915

11991012

1199119

+11990915

1199109

11991112

+ 1199099

11991012

11991115

+ 1199099

11991015

11991112

)

+ 3924756 (11990912

11991018

1199116

+ 11990912

1199106

11991118

+ 11990918

11991012

1199116

+11990918

1199106

11991112

+ 1199096

11991012

11991118

+ 1199096

11991018

11991112

)

+ 58344 (11990912

11991021

1199113

+ 11990912

1199103

11991121

+ 11990921

11991012

1199113

+11990921

1199103

11991112

+ 1199093

11991012

11991121

+ 1199093

11991021

11991112

)

+ 102 (11990912

11991024

+ 11990912

11991124

+ 11990924

11991012

+11990924

11991112

+ 11991012

11991124

+ 11991024

11991112

)

+ 170544 (11990915

11991018

1199113

+ 11990915

1199103

11991118

+ 11990918

11991015

1199113

+11990918

1199103

11991115

+ 1199093

11991015

11991118

+ 1199093

11991018

11991115

)

+ 641784 (11990921

1199106

1199119

+ 11990921

1199109

1199116

+ 1199096

11991021

1199119

+1199096

1199109

11991121

+ 1199099

11991021

1199116

+ 1199099

1199106

11991121

)

+ 6732 (11990924

1199103

1199119

+ 11990924

1199109

1199113

+ 1199093

11991024

1199119

+1199093

1199109

11991124

+ 1199099

11991024

1199113

+ 1199099

1199103

11991124

)

(18)

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

International Journal of Combinatorics 3

Proof The weight enumerator of 119861 is written as

119882119861(119910) = 1 + 119886119910

16

+ 11988711991020

+ 11988811991024

+ 11988911991028

+ 11989011991032

+ (27

minus 1 minus 119886 minus 119887 minus 119888 minus 119889 minus 119890) 11991036

(7)

where 119886 119887 119888 119889 and 119890 are nonnegative integers By theMacWilliams identity the weight enumerator of 119861perp is givenby

119882119861perp (119910) = 1 +

1

16(minus567 + 5119886 + 4119887 + 3119888 + 2119889 + 119890) 119910

+1

2(1260 minus 10119886 minus 10119887 minus 9119888 minus 7119889 minus 4119890) 119910

2

+1

16(minus112455 + 885119886 + 900119887

+ 883119888 + 770119889 + 497119890) 1199103

+ sdot sdot sdot

(8)

Since 119889(119861perp

) ge 4 the weight enumerator of 119861 is written using119886 and 119887

119882119861(119910) = 1 + 119886119910

16

+ 11988711991020

+ (882 minus 10119886 minus 4119887) 11991024

+ (minus1638 + 20119886 + 6119887) 11991028

+ (1197 minus 15119886 minus 4119887) 11991032

+ (minus314 + 4119886 + 119887) 11991036

(9)

Suppose that119861 does not contain the all-one vector 1Then119887 = 314minus4119886 In this case since the coefficients of 11991024 and 119910

28

are minus374 + 6119886 and 246 minus 4119886 these yield that 119886 ge 62 and 119886 le

61 respectively which gives the contradiction Hence 119861

contains 1 Then 119887 = 315 minus 4119886 Since the coefficient 119886 minus 63 of11991032 is 0 the weight enumerator of119861 is uniquely determined as

1 + 6311991016

+ 6311991020

+ 11991036

Remark 3 A similar approach shows that the weight enu-merator of a binary doubly even [36 8] code 119861 satisfyingconditions (1) is uniquely determined as 1+ 153119910

16

+7211991020

+

3011991024

It was shown in [8] that there are four inequivalent binary[36 7 16] codes containing 1The four codes are doubly evenHence there are exactly four binary doubly even [36 7] codessatisfying conditions (1) up to equivalenceThe four codes areoptimal in the sense that these codes achieve theGray-Rankinbound and the codewords of weight 16 are corresponding toquasi-symmetric SDP 2-(36 16 12) designs [15] Let 119861

36119894be

the binary doubly even [36 7 16] code corresponding to thequasi-symmetric SDP 2-(36 16 12) design which is theresidual design of the symmetric SDP 2-(64 28 12) design119863

119894

in [8 Section 5] (119894 = 1 2 3 4) As described above all self-dual Z

4-codes 119862 with 119862

(1)

= 11986136119894

have 119889119864(119862) = 16 (119894 =

1 2 3 4) Hence 1198604(119862) are extremal

Using the method in [13 Section 3] self-dual Z4-codes

119862 are constructed from 11986136119894

Then ten extremal unimodularlattices 119860

4(11986236119894

) (119894 = 1 2 10) are constructed where

119862(1)

36119894= 119861362

(119894 = 1 2 3) 119862(1)36119894

= 119861363

(119894 = 4 5 6 7) and119862(1)

36119894= 119861364

(119894 = 8 9 10) To distinguish between the knownlattices and our lattices we give in Table 1 the kissing numbers120591(119871) 119899

1(119871) 1198992(119871) and the orders Aut(119871) of the automor-

phism groups where 119899119894(119871) denotes the number of vectors of

norm 3 in 119873119890119894(119871) defined in (2) (119894 = 1 2) These have been

calculated by Magma Table 1 shows the following

Lemma 4 The five known lattices and the ten extremal uni-modular lattices119860

4(11986236119894

) (119894 = 1 2 10) are nonisomorphicto each other

Remark 5 In this way we have found two more extremalunimodular lattices 119860

4(119862) where 119862 are self-dual Z

4-codes

with 119862(1)

= 119861361

However we have verified by Magma thatthe two lattices are isomorphic to 119873

36in [3] and 119860

4(11986236) in

[4]

Remark 6 For 119871 = 1198604(11986236119894

) (119894 = 1 2 10) it followsfrom 120591(119871) and 119899

1(119871) 1198992(119871) that one of the two unimodular

neighbors 1198731198901(119871) and 119873119890

2(119871) defined in (2) is extremal We

have verified by Magma that the extremal one is isomorphicto 1198604(11986236119894

)

For 119894 = 1 2 10 the code 11986236119894

is equivalent to somecode 119862

36119894with generator matrix of the form

(1198687

119860 1198611+ 21198612

119874 211986822

2119863) (10)

where 119860 1198611 1198612 and 119863 are (1 0)-matrices 119868

119899denotes the

identity matrix of order 119899 and 119874 denotes the 22 times 7 zeromatrix We only list in (11) the 7 times 29 matrix 119872

119894=

(119860 1198611+ 21198612) to save space Note that (119874 2119868

222119863) in (10)

can be obtained from (1198687

119860 1198611+ 21198612) since 119862

36119894

(2)

=

11986236119894

(1)perp

A generatormatrix of1198604(11986236119894

) is obtained from thatof 11986236119894

1198721=

(((

(

01101111111000000110000333322

10101101001000111110011210223

11000011001110100111100010321

11010100101011010011012032011

00110101010101001101013312302

00000000000001111111112111333

00011111111110000001111213133

)))

)

1198722=

(((

(

01101111111000000110002111100

10101101001000111110011210023

11000011001110100111100010123

11010100101011010011012032211

00110101010101001101013312300

00000000000001111111112113131

00011111111110000001111211331

)))

)

4 International Journal of Combinatorics

Table 1 Extremal unimodular lattices in dimension 36

Lattices 119871 120591(119871) 1198991(119871) 119899

2(119871) Aut(119871)

Sp4(4)D84 in [1] 42840 480 480 3133440011986636in [2 Table 2] 42840 144 816 576

11987336in [3] 42840 0 960 849346560

1198604(11986236) in [4] 51032 0 840 660602880

1198606(119862366

(11986318)) in [5] 42840 384 576 288

1198604(119862361

) in Section 3 51032 0 840 62914561198604(119862362

) in Section 3 42840 0 960 62914561198604(119862363

) in Section 3 51032 0 840 220200961198604(119862364

) in Section 3 51032 0 840 19660801198604(119862365

) in Section 3 51032 0 840 15728641198604(119862366

) in Section 3 42840 0 960 26214401198604(119862367

) in Section 3 42840 0 960 19660801198604(119862368

) in Section 3 42840 0 960 3932161198604(119862369

) in Section 3 51032 0 840 13762561198604(1198623610

) in Section 3 51032 0 840 3932161198605(119863361

) in Section 4 42840 144 816 1441198605(119863362

) in Section 4 42840 456 504 721198606(119863363

) in Section 4 42840 240 720 2881198606(119863364

) in Section 4 42840 240 720 5761198607(119863365

) in Section 4 42840 288 672 2881198607(119863366

) in Section 4 42840 144 816 721198607(119863367

) in Section 4 42840 144 816 2881198609(119863368

) in Section 4 42840 384 576 14411986019(119863369

) in Section 4 42840 288 672 1441198605(119864361

) in Section 4 42840 456 504 721198606(119864362

) in Section 4 42840 384 576 144

1198723=

(((

(

01101111111000000110002131300

10101101001000111110011210223

11000011001110100111100010121

11010100101011010011012032011

00110101010101001101013310102

00000000000001111111112131131

00011111111110000001111233331

)))

)

1198724=

(((

(

01101111111001111101100220313

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100021110

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198725=

(((

(

01101111111001111101102202133

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100221310

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198726=

(((

(

01101111111001111101102000313

10101101001001001110101231032

11000011001110110100102101103

00111001100011100101113210223

11011000011101100011100021110

00000000000001111111112313113

00011111111110000001113103333

)))

)

1198727=

(((

(

01101111111001111101102202313

10101101001001001110101231030

11000011001110110100102101121

00111001100011100101113210201

11011000011101100011100021112

00000000000001111111112111113

00011111111110000001113301333

)))

)

1198728=

(((

(

11111110000111111111000022313

11110001100111000000001130313

01000110111001110100011231021

10011100001011100001101233011

01010101010010101101003301321

00000000011111111111112331111

01111111100000000111111233313

)))

)

International Journal of Combinatorics 5

1198729=

(((

(

11111110000111111111002200111

11110001100111000000001332311

01000110111001110100013211021

10011100001011100001103213011

01010101010010101101000331321

01111111100000000111111211113

00000000011111111111111303313

)))

)

11987210

=

(((

(

11111110000111111111002202313

11110001100111000000001130311

01000110111001110100011211021

10011100001011100001101213011

01010101010010101101003321123

00000000011111111111112313313

01111111100000000111111211111

)))

)

(11)

32 Unimodular Lattices with Long Shadows The possibletheta series of a unimodular lattice 119871 in dimension 36 havingminimum norm 3 and its shadow are as follows

1 + (960 minus 120572) 1199023

+ (42840 + 4096120573) 1199024

+ sdot sdot sdot

120573119902 + (120572 minus 60120573) 1199023

+ (3833856 minus 36120572 + 1734120573) 1199025

+ sdot sdot sdot

(12)

respectively where 120572 and 120573 are integers with 0 le 120573 le 12057260 lt

16 [3] Then the kissing number of 119871 is at most 960 andmin(119878(119871)) le 5 Unimodular lattices 119871 with min(119871) = 3 andmin(119878(119871)) = 5 are often called unimodular lattices with longshadows (see [9]) Only one unimodular lattice 119871 in dimen-sion 36 with min(119871) = 3 and min(119878(119871)) = 5 was knownnamely 119860

4(11986236) in [3]

Let 119871 be one of 1198604(119862362

) 1198604(119862366

) 1198604(119862367

) and1198604(119862368

) Since 1198991(119871) 1198992(119871) = 0 960 one of the two

unimodular neighbors 1198731198901(119871) and 119873119890

2(119871) in (2) is extremal

and the other is a unimodular lattice 1198711015840 with minimum norm

3 having shadow of minimum norm 5 We denote such lat-tices 1198711015840 constructed from119860

4(119862362

)1198604(119862366

)1198604(119862367

) and1198604(119862368

) by119873361

119873362

119873363

and119873364

respectivelyWe listin Table 2 the orders Aut(119873

36119894) (119894 = 1 2 3 4) of the auto-

morphism groups which have been calculated by MagmaTable 2 shows the following

Proposition7 There are at least 5 nonisomorphic unimodularlattices 119871 in dimension 36 withmin(119871) = 3 andmin(119878(119871)) = 5

4 From Self-Dual Z119896-Codes (119896 ge 5)

In this section we construct more extremal unimodularlattices in dimension 36 from self-dual Z

119896-codes (119896 ge 5)

Let119860119879 denote the transpose of amatrix119860 An 119899times119899matrixis negacirculant if it has the following form

(

1199030

1199031

sdot sdot sdot 119903119899minus1

minus119903119899minus1

1199030

sdot sdot sdot 119903119899minus2

minus119903119899minus2

minus119903119899minus1

sdot sdot sdot 119903119899minus3

minus1199031

minus1199032

sdot sdot sdot 1199030

) (13)

Table 2 Aut(11987336119894

) (119894 = 1 2 3 4)

Lattices 119871 Aut(119871)1198604(11986236) in [3] 1698693120

119873361

12582912119873362

5242880119873363

3932160119873364

786432

Let 11986336119894

(119894 = 1 2 9) and 11986436119894

(119894 = 1 2) be Z119896-codes of

length 36 with generator matrices of the following form

(11986818

119860 119861

minus119861119879

119860119879 ) (14)

where 119896 are listed in Table 3 and 119860 and 119861 are 9 times 9

negacirculant matrices with first rows 119903119860and 119903

119861listed in

Table 3 It is easy to see that these codes are self-dual since119860119860119879

+ 119861119861119879

= minus1198689 Thus 119860

119896(11986336119894

) (119894 = 1 2 9) and119860119896(11986436119894

) (119894 = 1 2) are unimodular lattices for 119896 given inTable 3 In addition we have verified by Magma that theselattices are extremal

To distinguish between the above eleven lattices and the15 known lattices in Table 1 we give 120591(119871) 119899

1(119871) 1198992(119871)

and Aut(119871) which have been calculated by MagmaThe two lattices have the identical (120591(119871) 119899

1(119871) 1198992(119871)

Aut(119871)) for each of the pairs (1198605(119864361

) 1198605(119863362

)) and(1198606(119864362

) 1198609(119863368

)) However we have verified by Magmathat the two lattices are nonisomorphic for each pair There-fore we have the following

Lemma 8 The 26 lattices in Table 1 are nonisomorphic to eachother

Lemma 8 establishes Proposition 1

Remark 9 Similar to Remark 6 it is known [3] that theextremal neighbor is isomorphic to 119871 for the case where 119871 is11987336in [3] and we have verified by Magma that the extremal

neighbor is isomorphic to 119871 for the case where 119871 is 1198604(11986236)

in [4]

5 Related Ternary Self-Dual Codes

In this section we give a certain short observation on ternaryself-dual codes related to some extremal odd unimodularlattices in dimension 36

51 Unimodular Lattices from Ternary Self-Dual Codes Let11987936

be a ternary self-dual code of length 36 The two uni-modular neighbors 119873119890

1(1198603(11987936)) and 119873119890

2(1198603(11987936)) given

in (2) are described in [16] as 119871119878(11987936) and 119871

119879(11987936) In this

section we use the notation 119871119878(11987936) and 119871

119879(11987936) instead of

1198731198901(1198603(11987936)) and 119873119890

2(1198603(11987936)) since the explicit construc-

tions and some properties of 119871119878(11987936) and 119871

119879(11987936) are given in

[16] By Theorem 6 in [16] (see also Theorem 31 in [2])119871119879(11987936) is extremal when119879

36satisfies the following condition

6 International Journal of Combinatorics

Table 3 Self-dual Z119896-codes of length 36

Codes 119896 119903119860

119903119861

119863361

5 (0 0 0 1 2 2 0 4 2) (0 0 0 0 4 3 3 0 1)

119863362

5 (0 0 0 1 3 0 2 0 4) (3 0 4 1 4 0 0 4 4)

119863363

6 (0 1 5 3 2 0 3 5 1) (3 1 0 0 5 1 1 1 1)

119863364

6 (0 1 3 5 1 5 5 4 4) (4 0 3 2 4 5 5 2 4)

119863365

7 (0 1 6 3 5 0 4 5 4) (0 1 6 3 5 2 1 5 1)

119863366

7 (0 1 1 3 2 6 1 4 6) (0 1 4 0 5 3 6 2 0)

119863367

7 (0 0 0 1 5 5 5 1 1) (0 5 4 2 5 1 1 3 6)

119863368

9 (0 0 0 1 0 4 3 0 0) (0 4 1 5 3 5 1 7 0)

119863369

19 (0 0 0 1 15 15 9 6 5) (14 16 0 14 15 8 8 3 12)

119864361

5 (0 1 0 2 1 3 2 2 0) (2 0 1 0 1 1 2 3 1)

119864362

6 (0 1 5 3 4 4 1 1 0) (4 0 1 3 4 2 3 0 1)

(a) and both 119871119878(11987936) and 119871

119879(11987936) are extremal when 119879

36

satisfies the following condition (b)(a) extremal (minimum weight 12) and admissible (the

number of 1rsquos in the components of every codewordof weight 36 is even)

(b) minimum weight 9 and maximum weight 33For each of (a) and (b) no ternary self-dual code satisfyingthe condition is currently known

52 Condition (a) Suppose that 11987936

satisfies condition (a)Since 119879

36has minimum weight 12 119860

3(11987936) has minimum

norm 3 and kissing number 72 By Theorem 6 in [16]min(119871

119879(11987936)) = 4 and min(119871

119878(11987936)) = 3 Hence since the

shadow of 119871119879(11987936) contains no vector of norm 1 by (3) and

(4) 119871119879(11987936) has theta series 1+ 42840119902

4

+ 19169281199025

+ sdot sdot sdot Itfollows that 119899

1(119871119879(11987936)) 1198992(119871119879(11987936)) = 72 888

By Theorem 1 in [17] the possible complete weightenumerator 119882

119862(119909 119910 119911) of a ternary extremal self-dual code

119862 of length 36 containing 1 is written as

119886112057536

+ 11988621205723

12+ 11988631205722

121205732

6+ 119886412057212

(1205732

6)2

+ 1198865(1205732

6)3

+ 119886612057361205741812057212

+ 11988671205736120574181205732

6

(15)

using some 119886119894isin R (119894 = 1 2 7) where 120572

12= 119886(119886

3

+ 81199013

)1205736= 1198862

minus12119887 12057418

= 1198866

minus201198863

1199013

minus81199016 and 120575

36= 1199013

(1198863

minus1199013

)3

and 119886 = 1199093

+ 1199103

+ 1199113 119901 = 3119909119910119911 and 119887 = 119909

3

1199103

+ 1199093

1199113

+ 1199103

1199113

From the minimum weight we have the following

1198862=

3281

13824minus

1198861

64 119886

3=

203

4608minus

91198861

256

1198864=

1763

13824+

31198861

128 119886

5= minus

277

13824minus

1198861

256

1198866=

1133

1728+

31198861

64 119886

7= minus

77

1728minus

1198861

64

(16)

Since 119882119862(119909 119910 119911) contains the term (15180 + 2916119886

1)11991015

11991121

if 119862 is admissible then

1198861= minus

15180

2916 (17)

Hence the complete weight enumerator of a ternary admis-sible extremal self-dual code containing 1 is uniquely deter-mined which is listed in (18)

11990936

+ 11991036

+ 11991136

+ 7870626011990912

11991012

11991112

+ 682 (11990918

11991018

+ 11990918

11991118

+ 11991018

11991118

)

+ 7019232 (11990915

11991015

1199116

+ 11990915

1199106

11991115

+ 1199096

11991015

11991115

)

+ 29172 (11990924

1199106

1199116

+ 1199096

11991024

1199116

+ 1199096

1199106

11991124

)

+ 10260316 (11990918

1199109

1199119

+ 1199099

11991018

1199119

+ 1199099

1199109

11991118

)

+ 37995408 (11990912

11991015

1199119

+ 11990912

1199109

11991115

+ 11990915

11991012

1199119

+11990915

1199109

11991112

+ 1199099

11991012

11991115

+ 1199099

11991015

11991112

)

+ 3924756 (11990912

11991018

1199116

+ 11990912

1199106

11991118

+ 11990918

11991012

1199116

+11990918

1199106

11991112

+ 1199096

11991012

11991118

+ 1199096

11991018

11991112

)

+ 58344 (11990912

11991021

1199113

+ 11990912

1199103

11991121

+ 11990921

11991012

1199113

+11990921

1199103

11991112

+ 1199093

11991012

11991121

+ 1199093

11991021

11991112

)

+ 102 (11990912

11991024

+ 11990912

11991124

+ 11990924

11991012

+11990924

11991112

+ 11991012

11991124

+ 11991024

11991112

)

+ 170544 (11990915

11991018

1199113

+ 11990915

1199103

11991118

+ 11990918

11991015

1199113

+11990918

1199103

11991115

+ 1199093

11991015

11991118

+ 1199093

11991018

11991115

)

+ 641784 (11990921

1199106

1199119

+ 11990921

1199109

1199116

+ 1199096

11991021

1199119

+1199096

1199109

11991121

+ 1199099

11991021

1199116

+ 1199099

1199106

11991121

)

+ 6732 (11990924

1199103

1199119

+ 11990924

1199109

1199113

+ 1199093

11991024

1199119

+1199093

1199109

11991124

+ 1199099

11991024

1199113

+ 1199099

1199103

11991124

)

(18)

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

4 International Journal of Combinatorics

Table 1 Extremal unimodular lattices in dimension 36

Lattices 119871 120591(119871) 1198991(119871) 119899

2(119871) Aut(119871)

Sp4(4)D84 in [1] 42840 480 480 3133440011986636in [2 Table 2] 42840 144 816 576

11987336in [3] 42840 0 960 849346560

1198604(11986236) in [4] 51032 0 840 660602880

1198606(119862366

(11986318)) in [5] 42840 384 576 288

1198604(119862361

) in Section 3 51032 0 840 62914561198604(119862362

) in Section 3 42840 0 960 62914561198604(119862363

) in Section 3 51032 0 840 220200961198604(119862364

) in Section 3 51032 0 840 19660801198604(119862365

) in Section 3 51032 0 840 15728641198604(119862366

) in Section 3 42840 0 960 26214401198604(119862367

) in Section 3 42840 0 960 19660801198604(119862368

) in Section 3 42840 0 960 3932161198604(119862369

) in Section 3 51032 0 840 13762561198604(1198623610

) in Section 3 51032 0 840 3932161198605(119863361

) in Section 4 42840 144 816 1441198605(119863362

) in Section 4 42840 456 504 721198606(119863363

) in Section 4 42840 240 720 2881198606(119863364

) in Section 4 42840 240 720 5761198607(119863365

) in Section 4 42840 288 672 2881198607(119863366

) in Section 4 42840 144 816 721198607(119863367

) in Section 4 42840 144 816 2881198609(119863368

) in Section 4 42840 384 576 14411986019(119863369

) in Section 4 42840 288 672 1441198605(119864361

) in Section 4 42840 456 504 721198606(119864362

) in Section 4 42840 384 576 144

1198723=

(((

(

01101111111000000110002131300

10101101001000111110011210223

11000011001110100111100010121

11010100101011010011012032011

00110101010101001101013310102

00000000000001111111112131131

00011111111110000001111233331

)))

)

1198724=

(((

(

01101111111001111101100220313

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100021110

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198725=

(((

(

01101111111001111101102202133

10101101001001001110101231032

11000011001110110100102101101

00111001100011100101113010223

11011000011101100011100221310

00000000000001111111112111131

00011111111110000001113301311

)))

)

1198726=

(((

(

01101111111001111101102000313

10101101001001001110101231032

11000011001110110100102101103

00111001100011100101113210223

11011000011101100011100021110

00000000000001111111112313113

00011111111110000001113103333

)))

)

1198727=

(((

(

01101111111001111101102202313

10101101001001001110101231030

11000011001110110100102101121

00111001100011100101113210201

11011000011101100011100021112

00000000000001111111112111113

00011111111110000001113301333

)))

)

1198728=

(((

(

11111110000111111111000022313

11110001100111000000001130313

01000110111001110100011231021

10011100001011100001101233011

01010101010010101101003301321

00000000011111111111112331111

01111111100000000111111233313

)))

)

International Journal of Combinatorics 5

1198729=

(((

(

11111110000111111111002200111

11110001100111000000001332311

01000110111001110100013211021

10011100001011100001103213011

01010101010010101101000331321

01111111100000000111111211113

00000000011111111111111303313

)))

)

11987210

=

(((

(

11111110000111111111002202313

11110001100111000000001130311

01000110111001110100011211021

10011100001011100001101213011

01010101010010101101003321123

00000000011111111111112313313

01111111100000000111111211111

)))

)

(11)

32 Unimodular Lattices with Long Shadows The possibletheta series of a unimodular lattice 119871 in dimension 36 havingminimum norm 3 and its shadow are as follows

1 + (960 minus 120572) 1199023

+ (42840 + 4096120573) 1199024

+ sdot sdot sdot

120573119902 + (120572 minus 60120573) 1199023

+ (3833856 minus 36120572 + 1734120573) 1199025

+ sdot sdot sdot

(12)

respectively where 120572 and 120573 are integers with 0 le 120573 le 12057260 lt

16 [3] Then the kissing number of 119871 is at most 960 andmin(119878(119871)) le 5 Unimodular lattices 119871 with min(119871) = 3 andmin(119878(119871)) = 5 are often called unimodular lattices with longshadows (see [9]) Only one unimodular lattice 119871 in dimen-sion 36 with min(119871) = 3 and min(119878(119871)) = 5 was knownnamely 119860

4(11986236) in [3]

Let 119871 be one of 1198604(119862362

) 1198604(119862366

) 1198604(119862367

) and1198604(119862368

) Since 1198991(119871) 1198992(119871) = 0 960 one of the two

unimodular neighbors 1198731198901(119871) and 119873119890

2(119871) in (2) is extremal

and the other is a unimodular lattice 1198711015840 with minimum norm

3 having shadow of minimum norm 5 We denote such lat-tices 1198711015840 constructed from119860

4(119862362

)1198604(119862366

)1198604(119862367

) and1198604(119862368

) by119873361

119873362

119873363

and119873364

respectivelyWe listin Table 2 the orders Aut(119873

36119894) (119894 = 1 2 3 4) of the auto-

morphism groups which have been calculated by MagmaTable 2 shows the following

Proposition7 There are at least 5 nonisomorphic unimodularlattices 119871 in dimension 36 withmin(119871) = 3 andmin(119878(119871)) = 5

4 From Self-Dual Z119896-Codes (119896 ge 5)

In this section we construct more extremal unimodularlattices in dimension 36 from self-dual Z

119896-codes (119896 ge 5)

Let119860119879 denote the transpose of amatrix119860 An 119899times119899matrixis negacirculant if it has the following form

(

1199030

1199031

sdot sdot sdot 119903119899minus1

minus119903119899minus1

1199030

sdot sdot sdot 119903119899minus2

minus119903119899minus2

minus119903119899minus1

sdot sdot sdot 119903119899minus3

minus1199031

minus1199032

sdot sdot sdot 1199030

) (13)

Table 2 Aut(11987336119894

) (119894 = 1 2 3 4)

Lattices 119871 Aut(119871)1198604(11986236) in [3] 1698693120

119873361

12582912119873362

5242880119873363

3932160119873364

786432

Let 11986336119894

(119894 = 1 2 9) and 11986436119894

(119894 = 1 2) be Z119896-codes of

length 36 with generator matrices of the following form

(11986818

119860 119861

minus119861119879

119860119879 ) (14)

where 119896 are listed in Table 3 and 119860 and 119861 are 9 times 9

negacirculant matrices with first rows 119903119860and 119903

119861listed in

Table 3 It is easy to see that these codes are self-dual since119860119860119879

+ 119861119861119879

= minus1198689 Thus 119860

119896(11986336119894

) (119894 = 1 2 9) and119860119896(11986436119894

) (119894 = 1 2) are unimodular lattices for 119896 given inTable 3 In addition we have verified by Magma that theselattices are extremal

To distinguish between the above eleven lattices and the15 known lattices in Table 1 we give 120591(119871) 119899

1(119871) 1198992(119871)

and Aut(119871) which have been calculated by MagmaThe two lattices have the identical (120591(119871) 119899

1(119871) 1198992(119871)

Aut(119871)) for each of the pairs (1198605(119864361

) 1198605(119863362

)) and(1198606(119864362

) 1198609(119863368

)) However we have verified by Magmathat the two lattices are nonisomorphic for each pair There-fore we have the following

Lemma 8 The 26 lattices in Table 1 are nonisomorphic to eachother

Lemma 8 establishes Proposition 1

Remark 9 Similar to Remark 6 it is known [3] that theextremal neighbor is isomorphic to 119871 for the case where 119871 is11987336in [3] and we have verified by Magma that the extremal

neighbor is isomorphic to 119871 for the case where 119871 is 1198604(11986236)

in [4]

5 Related Ternary Self-Dual Codes

In this section we give a certain short observation on ternaryself-dual codes related to some extremal odd unimodularlattices in dimension 36

51 Unimodular Lattices from Ternary Self-Dual Codes Let11987936

be a ternary self-dual code of length 36 The two uni-modular neighbors 119873119890

1(1198603(11987936)) and 119873119890

2(1198603(11987936)) given

in (2) are described in [16] as 119871119878(11987936) and 119871

119879(11987936) In this

section we use the notation 119871119878(11987936) and 119871

119879(11987936) instead of

1198731198901(1198603(11987936)) and 119873119890

2(1198603(11987936)) since the explicit construc-

tions and some properties of 119871119878(11987936) and 119871

119879(11987936) are given in

[16] By Theorem 6 in [16] (see also Theorem 31 in [2])119871119879(11987936) is extremal when119879

36satisfies the following condition

6 International Journal of Combinatorics

Table 3 Self-dual Z119896-codes of length 36

Codes 119896 119903119860

119903119861

119863361

5 (0 0 0 1 2 2 0 4 2) (0 0 0 0 4 3 3 0 1)

119863362

5 (0 0 0 1 3 0 2 0 4) (3 0 4 1 4 0 0 4 4)

119863363

6 (0 1 5 3 2 0 3 5 1) (3 1 0 0 5 1 1 1 1)

119863364

6 (0 1 3 5 1 5 5 4 4) (4 0 3 2 4 5 5 2 4)

119863365

7 (0 1 6 3 5 0 4 5 4) (0 1 6 3 5 2 1 5 1)

119863366

7 (0 1 1 3 2 6 1 4 6) (0 1 4 0 5 3 6 2 0)

119863367

7 (0 0 0 1 5 5 5 1 1) (0 5 4 2 5 1 1 3 6)

119863368

9 (0 0 0 1 0 4 3 0 0) (0 4 1 5 3 5 1 7 0)

119863369

19 (0 0 0 1 15 15 9 6 5) (14 16 0 14 15 8 8 3 12)

119864361

5 (0 1 0 2 1 3 2 2 0) (2 0 1 0 1 1 2 3 1)

119864362

6 (0 1 5 3 4 4 1 1 0) (4 0 1 3 4 2 3 0 1)

(a) and both 119871119878(11987936) and 119871

119879(11987936) are extremal when 119879

36

satisfies the following condition (b)(a) extremal (minimum weight 12) and admissible (the

number of 1rsquos in the components of every codewordof weight 36 is even)

(b) minimum weight 9 and maximum weight 33For each of (a) and (b) no ternary self-dual code satisfyingthe condition is currently known

52 Condition (a) Suppose that 11987936

satisfies condition (a)Since 119879

36has minimum weight 12 119860

3(11987936) has minimum

norm 3 and kissing number 72 By Theorem 6 in [16]min(119871

119879(11987936)) = 4 and min(119871

119878(11987936)) = 3 Hence since the

shadow of 119871119879(11987936) contains no vector of norm 1 by (3) and

(4) 119871119879(11987936) has theta series 1+ 42840119902

4

+ 19169281199025

+ sdot sdot sdot Itfollows that 119899

1(119871119879(11987936)) 1198992(119871119879(11987936)) = 72 888

By Theorem 1 in [17] the possible complete weightenumerator 119882

119862(119909 119910 119911) of a ternary extremal self-dual code

119862 of length 36 containing 1 is written as

119886112057536

+ 11988621205723

12+ 11988631205722

121205732

6+ 119886412057212

(1205732

6)2

+ 1198865(1205732

6)3

+ 119886612057361205741812057212

+ 11988671205736120574181205732

6

(15)

using some 119886119894isin R (119894 = 1 2 7) where 120572

12= 119886(119886

3

+ 81199013

)1205736= 1198862

minus12119887 12057418

= 1198866

minus201198863

1199013

minus81199016 and 120575

36= 1199013

(1198863

minus1199013

)3

and 119886 = 1199093

+ 1199103

+ 1199113 119901 = 3119909119910119911 and 119887 = 119909

3

1199103

+ 1199093

1199113

+ 1199103

1199113

From the minimum weight we have the following

1198862=

3281

13824minus

1198861

64 119886

3=

203

4608minus

91198861

256

1198864=

1763

13824+

31198861

128 119886

5= minus

277

13824minus

1198861

256

1198866=

1133

1728+

31198861

64 119886

7= minus

77

1728minus

1198861

64

(16)

Since 119882119862(119909 119910 119911) contains the term (15180 + 2916119886

1)11991015

11991121

if 119862 is admissible then

1198861= minus

15180

2916 (17)

Hence the complete weight enumerator of a ternary admis-sible extremal self-dual code containing 1 is uniquely deter-mined which is listed in (18)

11990936

+ 11991036

+ 11991136

+ 7870626011990912

11991012

11991112

+ 682 (11990918

11991018

+ 11990918

11991118

+ 11991018

11991118

)

+ 7019232 (11990915

11991015

1199116

+ 11990915

1199106

11991115

+ 1199096

11991015

11991115

)

+ 29172 (11990924

1199106

1199116

+ 1199096

11991024

1199116

+ 1199096

1199106

11991124

)

+ 10260316 (11990918

1199109

1199119

+ 1199099

11991018

1199119

+ 1199099

1199109

11991118

)

+ 37995408 (11990912

11991015

1199119

+ 11990912

1199109

11991115

+ 11990915

11991012

1199119

+11990915

1199109

11991112

+ 1199099

11991012

11991115

+ 1199099

11991015

11991112

)

+ 3924756 (11990912

11991018

1199116

+ 11990912

1199106

11991118

+ 11990918

11991012

1199116

+11990918

1199106

11991112

+ 1199096

11991012

11991118

+ 1199096

11991018

11991112

)

+ 58344 (11990912

11991021

1199113

+ 11990912

1199103

11991121

+ 11990921

11991012

1199113

+11990921

1199103

11991112

+ 1199093

11991012

11991121

+ 1199093

11991021

11991112

)

+ 102 (11990912

11991024

+ 11990912

11991124

+ 11990924

11991012

+11990924

11991112

+ 11991012

11991124

+ 11991024

11991112

)

+ 170544 (11990915

11991018

1199113

+ 11990915

1199103

11991118

+ 11990918

11991015

1199113

+11990918

1199103

11991115

+ 1199093

11991015

11991118

+ 1199093

11991018

11991115

)

+ 641784 (11990921

1199106

1199119

+ 11990921

1199109

1199116

+ 1199096

11991021

1199119

+1199096

1199109

11991121

+ 1199099

11991021

1199116

+ 1199099

1199106

11991121

)

+ 6732 (11990924

1199103

1199119

+ 11990924

1199109

1199113

+ 1199093

11991024

1199119

+1199093

1199109

11991124

+ 1199099

11991024

1199113

+ 1199099

1199103

11991124

)

(18)

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

International Journal of Combinatorics 5

1198729=

(((

(

11111110000111111111002200111

11110001100111000000001332311

01000110111001110100013211021

10011100001011100001103213011

01010101010010101101000331321

01111111100000000111111211113

00000000011111111111111303313

)))

)

11987210

=

(((

(

11111110000111111111002202313

11110001100111000000001130311

01000110111001110100011211021

10011100001011100001101213011

01010101010010101101003321123

00000000011111111111112313313

01111111100000000111111211111

)))

)

(11)

32 Unimodular Lattices with Long Shadows The possibletheta series of a unimodular lattice 119871 in dimension 36 havingminimum norm 3 and its shadow are as follows

1 + (960 minus 120572) 1199023

+ (42840 + 4096120573) 1199024

+ sdot sdot sdot

120573119902 + (120572 minus 60120573) 1199023

+ (3833856 minus 36120572 + 1734120573) 1199025

+ sdot sdot sdot

(12)

respectively where 120572 and 120573 are integers with 0 le 120573 le 12057260 lt

16 [3] Then the kissing number of 119871 is at most 960 andmin(119878(119871)) le 5 Unimodular lattices 119871 with min(119871) = 3 andmin(119878(119871)) = 5 are often called unimodular lattices with longshadows (see [9]) Only one unimodular lattice 119871 in dimen-sion 36 with min(119871) = 3 and min(119878(119871)) = 5 was knownnamely 119860

4(11986236) in [3]

Let 119871 be one of 1198604(119862362

) 1198604(119862366

) 1198604(119862367

) and1198604(119862368

) Since 1198991(119871) 1198992(119871) = 0 960 one of the two

unimodular neighbors 1198731198901(119871) and 119873119890

2(119871) in (2) is extremal

and the other is a unimodular lattice 1198711015840 with minimum norm

3 having shadow of minimum norm 5 We denote such lat-tices 1198711015840 constructed from119860

4(119862362

)1198604(119862366

)1198604(119862367

) and1198604(119862368

) by119873361

119873362

119873363

and119873364

respectivelyWe listin Table 2 the orders Aut(119873

36119894) (119894 = 1 2 3 4) of the auto-

morphism groups which have been calculated by MagmaTable 2 shows the following

Proposition7 There are at least 5 nonisomorphic unimodularlattices 119871 in dimension 36 withmin(119871) = 3 andmin(119878(119871)) = 5

4 From Self-Dual Z119896-Codes (119896 ge 5)

In this section we construct more extremal unimodularlattices in dimension 36 from self-dual Z

119896-codes (119896 ge 5)

Let119860119879 denote the transpose of amatrix119860 An 119899times119899matrixis negacirculant if it has the following form

(

1199030

1199031

sdot sdot sdot 119903119899minus1

minus119903119899minus1

1199030

sdot sdot sdot 119903119899minus2

minus119903119899minus2

minus119903119899minus1

sdot sdot sdot 119903119899minus3

minus1199031

minus1199032

sdot sdot sdot 1199030

) (13)

Table 2 Aut(11987336119894

) (119894 = 1 2 3 4)

Lattices 119871 Aut(119871)1198604(11986236) in [3] 1698693120

119873361

12582912119873362

5242880119873363

3932160119873364

786432

Let 11986336119894

(119894 = 1 2 9) and 11986436119894

(119894 = 1 2) be Z119896-codes of

length 36 with generator matrices of the following form

(11986818

119860 119861

minus119861119879

119860119879 ) (14)

where 119896 are listed in Table 3 and 119860 and 119861 are 9 times 9

negacirculant matrices with first rows 119903119860and 119903

119861listed in

Table 3 It is easy to see that these codes are self-dual since119860119860119879

+ 119861119861119879

= minus1198689 Thus 119860

119896(11986336119894

) (119894 = 1 2 9) and119860119896(11986436119894

) (119894 = 1 2) are unimodular lattices for 119896 given inTable 3 In addition we have verified by Magma that theselattices are extremal

To distinguish between the above eleven lattices and the15 known lattices in Table 1 we give 120591(119871) 119899

1(119871) 1198992(119871)

and Aut(119871) which have been calculated by MagmaThe two lattices have the identical (120591(119871) 119899

1(119871) 1198992(119871)

Aut(119871)) for each of the pairs (1198605(119864361

) 1198605(119863362

)) and(1198606(119864362

) 1198609(119863368

)) However we have verified by Magmathat the two lattices are nonisomorphic for each pair There-fore we have the following

Lemma 8 The 26 lattices in Table 1 are nonisomorphic to eachother

Lemma 8 establishes Proposition 1

Remark 9 Similar to Remark 6 it is known [3] that theextremal neighbor is isomorphic to 119871 for the case where 119871 is11987336in [3] and we have verified by Magma that the extremal

neighbor is isomorphic to 119871 for the case where 119871 is 1198604(11986236)

in [4]

5 Related Ternary Self-Dual Codes

In this section we give a certain short observation on ternaryself-dual codes related to some extremal odd unimodularlattices in dimension 36

51 Unimodular Lattices from Ternary Self-Dual Codes Let11987936

be a ternary self-dual code of length 36 The two uni-modular neighbors 119873119890

1(1198603(11987936)) and 119873119890

2(1198603(11987936)) given

in (2) are described in [16] as 119871119878(11987936) and 119871

119879(11987936) In this

section we use the notation 119871119878(11987936) and 119871

119879(11987936) instead of

1198731198901(1198603(11987936)) and 119873119890

2(1198603(11987936)) since the explicit construc-

tions and some properties of 119871119878(11987936) and 119871

119879(11987936) are given in

[16] By Theorem 6 in [16] (see also Theorem 31 in [2])119871119879(11987936) is extremal when119879

36satisfies the following condition

6 International Journal of Combinatorics

Table 3 Self-dual Z119896-codes of length 36

Codes 119896 119903119860

119903119861

119863361

5 (0 0 0 1 2 2 0 4 2) (0 0 0 0 4 3 3 0 1)

119863362

5 (0 0 0 1 3 0 2 0 4) (3 0 4 1 4 0 0 4 4)

119863363

6 (0 1 5 3 2 0 3 5 1) (3 1 0 0 5 1 1 1 1)

119863364

6 (0 1 3 5 1 5 5 4 4) (4 0 3 2 4 5 5 2 4)

119863365

7 (0 1 6 3 5 0 4 5 4) (0 1 6 3 5 2 1 5 1)

119863366

7 (0 1 1 3 2 6 1 4 6) (0 1 4 0 5 3 6 2 0)

119863367

7 (0 0 0 1 5 5 5 1 1) (0 5 4 2 5 1 1 3 6)

119863368

9 (0 0 0 1 0 4 3 0 0) (0 4 1 5 3 5 1 7 0)

119863369

19 (0 0 0 1 15 15 9 6 5) (14 16 0 14 15 8 8 3 12)

119864361

5 (0 1 0 2 1 3 2 2 0) (2 0 1 0 1 1 2 3 1)

119864362

6 (0 1 5 3 4 4 1 1 0) (4 0 1 3 4 2 3 0 1)

(a) and both 119871119878(11987936) and 119871

119879(11987936) are extremal when 119879

36

satisfies the following condition (b)(a) extremal (minimum weight 12) and admissible (the

number of 1rsquos in the components of every codewordof weight 36 is even)

(b) minimum weight 9 and maximum weight 33For each of (a) and (b) no ternary self-dual code satisfyingthe condition is currently known

52 Condition (a) Suppose that 11987936

satisfies condition (a)Since 119879

36has minimum weight 12 119860

3(11987936) has minimum

norm 3 and kissing number 72 By Theorem 6 in [16]min(119871

119879(11987936)) = 4 and min(119871

119878(11987936)) = 3 Hence since the

shadow of 119871119879(11987936) contains no vector of norm 1 by (3) and

(4) 119871119879(11987936) has theta series 1+ 42840119902

4

+ 19169281199025

+ sdot sdot sdot Itfollows that 119899

1(119871119879(11987936)) 1198992(119871119879(11987936)) = 72 888

By Theorem 1 in [17] the possible complete weightenumerator 119882

119862(119909 119910 119911) of a ternary extremal self-dual code

119862 of length 36 containing 1 is written as

119886112057536

+ 11988621205723

12+ 11988631205722

121205732

6+ 119886412057212

(1205732

6)2

+ 1198865(1205732

6)3

+ 119886612057361205741812057212

+ 11988671205736120574181205732

6

(15)

using some 119886119894isin R (119894 = 1 2 7) where 120572

12= 119886(119886

3

+ 81199013

)1205736= 1198862

minus12119887 12057418

= 1198866

minus201198863

1199013

minus81199016 and 120575

36= 1199013

(1198863

minus1199013

)3

and 119886 = 1199093

+ 1199103

+ 1199113 119901 = 3119909119910119911 and 119887 = 119909

3

1199103

+ 1199093

1199113

+ 1199103

1199113

From the minimum weight we have the following

1198862=

3281

13824minus

1198861

64 119886

3=

203

4608minus

91198861

256

1198864=

1763

13824+

31198861

128 119886

5= minus

277

13824minus

1198861

256

1198866=

1133

1728+

31198861

64 119886

7= minus

77

1728minus

1198861

64

(16)

Since 119882119862(119909 119910 119911) contains the term (15180 + 2916119886

1)11991015

11991121

if 119862 is admissible then

1198861= minus

15180

2916 (17)

Hence the complete weight enumerator of a ternary admis-sible extremal self-dual code containing 1 is uniquely deter-mined which is listed in (18)

11990936

+ 11991036

+ 11991136

+ 7870626011990912

11991012

11991112

+ 682 (11990918

11991018

+ 11990918

11991118

+ 11991018

11991118

)

+ 7019232 (11990915

11991015

1199116

+ 11990915

1199106

11991115

+ 1199096

11991015

11991115

)

+ 29172 (11990924

1199106

1199116

+ 1199096

11991024

1199116

+ 1199096

1199106

11991124

)

+ 10260316 (11990918

1199109

1199119

+ 1199099

11991018

1199119

+ 1199099

1199109

11991118

)

+ 37995408 (11990912

11991015

1199119

+ 11990912

1199109

11991115

+ 11990915

11991012

1199119

+11990915

1199109

11991112

+ 1199099

11991012

11991115

+ 1199099

11991015

11991112

)

+ 3924756 (11990912

11991018

1199116

+ 11990912

1199106

11991118

+ 11990918

11991012

1199116

+11990918

1199106

11991112

+ 1199096

11991012

11991118

+ 1199096

11991018

11991112

)

+ 58344 (11990912

11991021

1199113

+ 11990912

1199103

11991121

+ 11990921

11991012

1199113

+11990921

1199103

11991112

+ 1199093

11991012

11991121

+ 1199093

11991021

11991112

)

+ 102 (11990912

11991024

+ 11990912

11991124

+ 11990924

11991012

+11990924

11991112

+ 11991012

11991124

+ 11991024

11991112

)

+ 170544 (11990915

11991018

1199113

+ 11990915

1199103

11991118

+ 11990918

11991015

1199113

+11990918

1199103

11991115

+ 1199093

11991015

11991118

+ 1199093

11991018

11991115

)

+ 641784 (11990921

1199106

1199119

+ 11990921

1199109

1199116

+ 1199096

11991021

1199119

+1199096

1199109

11991121

+ 1199099

11991021

1199116

+ 1199099

1199106

11991121

)

+ 6732 (11990924

1199103

1199119

+ 11990924

1199109

1199113

+ 1199093

11991024

1199119

+1199093

1199109

11991124

+ 1199099

11991024

1199113

+ 1199099

1199103

11991124

)

(18)

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

6 International Journal of Combinatorics

Table 3 Self-dual Z119896-codes of length 36

Codes 119896 119903119860

119903119861

119863361

5 (0 0 0 1 2 2 0 4 2) (0 0 0 0 4 3 3 0 1)

119863362

5 (0 0 0 1 3 0 2 0 4) (3 0 4 1 4 0 0 4 4)

119863363

6 (0 1 5 3 2 0 3 5 1) (3 1 0 0 5 1 1 1 1)

119863364

6 (0 1 3 5 1 5 5 4 4) (4 0 3 2 4 5 5 2 4)

119863365

7 (0 1 6 3 5 0 4 5 4) (0 1 6 3 5 2 1 5 1)

119863366

7 (0 1 1 3 2 6 1 4 6) (0 1 4 0 5 3 6 2 0)

119863367

7 (0 0 0 1 5 5 5 1 1) (0 5 4 2 5 1 1 3 6)

119863368

9 (0 0 0 1 0 4 3 0 0) (0 4 1 5 3 5 1 7 0)

119863369

19 (0 0 0 1 15 15 9 6 5) (14 16 0 14 15 8 8 3 12)

119864361

5 (0 1 0 2 1 3 2 2 0) (2 0 1 0 1 1 2 3 1)

119864362

6 (0 1 5 3 4 4 1 1 0) (4 0 1 3 4 2 3 0 1)

(a) and both 119871119878(11987936) and 119871

119879(11987936) are extremal when 119879

36

satisfies the following condition (b)(a) extremal (minimum weight 12) and admissible (the

number of 1rsquos in the components of every codewordof weight 36 is even)

(b) minimum weight 9 and maximum weight 33For each of (a) and (b) no ternary self-dual code satisfyingthe condition is currently known

52 Condition (a) Suppose that 11987936

satisfies condition (a)Since 119879

36has minimum weight 12 119860

3(11987936) has minimum

norm 3 and kissing number 72 By Theorem 6 in [16]min(119871

119879(11987936)) = 4 and min(119871

119878(11987936)) = 3 Hence since the

shadow of 119871119879(11987936) contains no vector of norm 1 by (3) and

(4) 119871119879(11987936) has theta series 1+ 42840119902

4

+ 19169281199025

+ sdot sdot sdot Itfollows that 119899

1(119871119879(11987936)) 1198992(119871119879(11987936)) = 72 888

By Theorem 1 in [17] the possible complete weightenumerator 119882

119862(119909 119910 119911) of a ternary extremal self-dual code

119862 of length 36 containing 1 is written as

119886112057536

+ 11988621205723

12+ 11988631205722

121205732

6+ 119886412057212

(1205732

6)2

+ 1198865(1205732

6)3

+ 119886612057361205741812057212

+ 11988671205736120574181205732

6

(15)

using some 119886119894isin R (119894 = 1 2 7) where 120572

12= 119886(119886

3

+ 81199013

)1205736= 1198862

minus12119887 12057418

= 1198866

minus201198863

1199013

minus81199016 and 120575

36= 1199013

(1198863

minus1199013

)3

and 119886 = 1199093

+ 1199103

+ 1199113 119901 = 3119909119910119911 and 119887 = 119909

3

1199103

+ 1199093

1199113

+ 1199103

1199113

From the minimum weight we have the following

1198862=

3281

13824minus

1198861

64 119886

3=

203

4608minus

91198861

256

1198864=

1763

13824+

31198861

128 119886

5= minus

277

13824minus

1198861

256

1198866=

1133

1728+

31198861

64 119886

7= minus

77

1728minus

1198861

64

(16)

Since 119882119862(119909 119910 119911) contains the term (15180 + 2916119886

1)11991015

11991121

if 119862 is admissible then

1198861= minus

15180

2916 (17)

Hence the complete weight enumerator of a ternary admis-sible extremal self-dual code containing 1 is uniquely deter-mined which is listed in (18)

11990936

+ 11991036

+ 11991136

+ 7870626011990912

11991012

11991112

+ 682 (11990918

11991018

+ 11990918

11991118

+ 11991018

11991118

)

+ 7019232 (11990915

11991015

1199116

+ 11990915

1199106

11991115

+ 1199096

11991015

11991115

)

+ 29172 (11990924

1199106

1199116

+ 1199096

11991024

1199116

+ 1199096

1199106

11991124

)

+ 10260316 (11990918

1199109

1199119

+ 1199099

11991018

1199119

+ 1199099

1199109

11991118

)

+ 37995408 (11990912

11991015

1199119

+ 11990912

1199109

11991115

+ 11990915

11991012

1199119

+11990915

1199109

11991112

+ 1199099

11991012

11991115

+ 1199099

11991015

11991112

)

+ 3924756 (11990912

11991018

1199116

+ 11990912

1199106

11991118

+ 11990918

11991012

1199116

+11990918

1199106

11991112

+ 1199096

11991012

11991118

+ 1199096

11991018

11991112

)

+ 58344 (11990912

11991021

1199113

+ 11990912

1199103

11991121

+ 11990921

11991012

1199113

+11990921

1199103

11991112

+ 1199093

11991012

11991121

+ 1199093

11991021

11991112

)

+ 102 (11990912

11991024

+ 11990912

11991124

+ 11990924

11991012

+11990924

11991112

+ 11991012

11991124

+ 11991024

11991112

)

+ 170544 (11990915

11991018

1199113

+ 11990915

1199103

11991118

+ 11990918

11991015

1199113

+11990918

1199103

11991115

+ 1199093

11991015

11991118

+ 1199093

11991018

11991115

)

+ 641784 (11990921

1199106

1199119

+ 11990921

1199109

1199116

+ 1199096

11991021

1199119

+1199096

1199109

11991121

+ 1199099

11991021

1199116

+ 1199099

1199106

11991121

)

+ 6732 (11990924

1199103

1199119

+ 11990924

1199109

1199113

+ 1199093

11991024

1199119

+1199093

1199109

11991124

+ 1199099

11991024

1199113

+ 1199099

1199103

11991124

)

(18)

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

International Journal of Combinatorics 7

53 Condition (b) Suppose that 11987936

satisfies condition (b)By the Gleason theorem (see Corollary 5 in [17]) the weightenumerator of 119879

36is uniquely determined as

1 + 8881199109

+ 3484811991012

+ 143222411991015

+ 1837768811991018

+ 9048225611991021

+ 16255159211991024

+ 9788307211991027

+ 1617868811991030

+ 47923211991033

(19)

By Theorem 6 in [16] (see also Theorem 31 in [2]) 119871119878(11987936)

and 119871119879(11987936) are extremal Hence min(119860

3(11987936)) = 3 and

min(119878(1198603(11987936))) = 5

Note that a unimodular lattice 119871 contains a 3-frame if andonly if 119871 cong 119860

3(119862) for some ternary self-dual code 119862 Let 119871

36

be any of the five lattices given in Table 2 Let 119871(3)36

be the set119909 minus119909 | (119909 119909) = 3 119909 isin 119871

36 We define the simple undi-

rected graph Γ(11987136) whose set of vertices is the set of 480pairs

in 119871(3)

36and two vertices 119909 minus119909 119910 minus119910 isin 119871

(3)

36are adjacent if

(119909 119910) = 0 It follows that the 3-frames in 11987136

are preciselythe 36-cliques in the graph Γ(119871

36) We have verified by

Magma that Γ(11987136) are regular graphs with valency 368 and

themaximum sizes of cliques in Γ(11987136) are 12 Hence none of

these lattices is constructed from some ternary self-dual codeby Construction A

Disclosure

This work was carried out at Yamagata University

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Masaaki Kitazume for bring-ing the observation in Section 5 to the authorrsquos attentionThiswork is supported by JSPS KAKENHI Grant no 23340021

References

[1] G Nebe andN J A Sloane ldquoUnimodular lattices together witha table of the best such lattices in A Catalogue of Latticesrdquohttpwwwmathrwth-aachendesimGabrieleNebeLATTICES

[2] P Gaborit ldquoConstruction of new extremal unimodular latticesrdquoEuropean Journal of Combinatorics vol 25 no 4 pp 549ndash5642004

[3] M Harada ldquoConstruction of some unimodular lattices withlong shadowsrdquo International Journal of Number Theory vol 7no 5 pp 1345ndash1358 2011

[4] M Harada ldquoOptimal self-dualZ4-codes and a unimodular lat-

tice in dimension 41rdquo Finite Fields andTheir Applications vol 18no 3 pp 529ndash536 2012

[5] M Harada ldquoExtremal Type II Z119896-codes and 119896-frames of odd

unimodular latticesrdquo To appear in IEEE Transactions on Infor-mation Theory

[6] E M Rains and N J Sloane ldquoThe shadow theory of modularand unimodular latticesrdquo Journal of NumberTheory vol 73 no2 pp 359ndash389 1998

[7] J H Conway and N J Sloane ldquoA note on optimal unimodularlatticesrdquo Journal of Number Theory vol 72 no 2 pp 357ndash3621998

[8] C Parker E Spence and V D Tonchev ldquoDesigns with the sym-metric difference property on 64 points and their groupsrdquoJournal of Combinatorial Theory Series A vol 67 no 1 pp 23ndash43 1994

[9] GNebe andBVenkov ldquoUnimodular latticeswith long shadowrdquoJournal of Number Theory vol 99 no 2 pp 307ndash317 2003

[10] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I the user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997

[11] J H Conway and N J A Sloane Sphere Packing Lattices andGroups Springer New York NY USA 3rd edition 1999

[12] J H Conway andN J Sloane ldquoSelf-dual codes over the integersmodulo 4rdquo Journal of CombinatorialTheory Series A vol 62 no1 pp 30ndash45 1993

[13] V Pless J S Leon and J Fields ldquoAll Z4codes of type II and

length 16 are knownrdquo Journal of Combinatorial Theory Series Avol 78 no 1 pp 32ndash50 1997

[14] A E Brouwer ldquoBounds on the size of linear codesrdquo inHandbookof Coding Theory V S Pless and W C Huffman Eds pp 295ndash461 Elsevier Amsterdam The Netherlands 1998

[15] D Jungnickel andVD Tonchev ldquoExponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankinboundrdquo Designs Codes and Cryptography vol 1 no 3 pp 247ndash253 1991

[16] M Harada M Kitazume and M Ozeki ldquoTernary code con-struction of unimodular lattices and self-dual codes over Z

6rdquo

Journal of Algebraic Combinatorics vol 16 no 2 pp 209ndash2232002

[17] C LMallows V Pless andN J A Sloane ldquoSelf-dual codes overGF(3)rdquo SIAM Journal on Applied Mathematics vol 31 pp 649ndash666 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Extremal Unimodular Lattices in Dimension 36downloads.hindawi.com/archive/2014/792471.pdfvation on ternary self-dual codes related to extremal odd unimodular lattices

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Extremal Laurent Polynomials

Extremal Laurent Polynomials

Chapter 2 Crystal Lattices and Reciprocal Lattices

Chapter 2 Crystal Lattices and Reciprocal Lattices

Calabi’s extremal K ahler metrics: An elementary introductionrodion/lib/P. Gauduchon. Calabi's extremal... · Admissible K ahler metrics 257 10.4. Extremal admissible K ahler metrics

Calabi’s extremal K ahler metrics: An elementary introductionrodion/lib/P. Gauduchon. Calabi's extremal... · Admissible K ahler metrics 257 10.4. Extremal admissible K ahler metrics

Updated 21 April2008 Linear Programs with Totally Unimodular Matrices.

Updated 21 April2008 Linear Programs with Totally Unimodular Matrices.

Automorphisms of even unimodular lattices and unramiï¬ed

Automorphisms of even unimodular lattices and unramiï¬ed

Pathophysiology of extremal states

Pathophysiology of extremal states

EXTREMAL QUANTILE REGRESSION - MITweb.mit.edu/~vchern/www/papers/c_extremal_QR.pdf · EXTREMAL QUANTILE REGRESSION VICTORCHERNOZHUKOV Abstract. Quantileregressionisanimportanttoolforestimationofconditionalquantilesofare-sponseY

EXTREMAL QUANTILE REGRESSION - MITweb.mit.edu/~vchern/www/papers/c_extremal_QR.pdf · EXTREMAL QUANTILE REGRESSION VICTORCHERNOZHUKOV Abstract. Quantileregressionisanimportanttoolforestimationofconditionalquantilesofare-sponseY

Lecture 6: Totally Unimodular Matrices · Unimodular Matrix • A unimodular matrix M is a square integer matrixwith determinant +1 or −1. • Equivalently, it is an integer matrix

Lecture 6: Totally Unimodular Matrices · Unimodular Matrix • A unimodular matrix M is a square integer matrixwith determinant +1 or −1. • Equivalently, it is an integer matrix

Totally Unimodular MatricesTotally Unimodular Matrices OR766 Theorem: A (0,+1,−1) matrix A is totally unimodular if both of the following conditions are satisfied: • Each column

Totally Unimodular MatricesTotally Unimodular Matrices OR766 Theorem: A (0,+1,−1) matrix A is totally unimodular if both of the following conditions are satisfied: • Each column

MSER (Maximally Stable Extremal Regions)

MSER (Maximally Stable Extremal Regions)

Point Lattices: Bravais Lattices - MIT - Massachusetts Institute

Point Lattices: Bravais Lattices - MIT - Massachusetts Institute

Graphic Lattices and Matrix Lattices Of Topological Coding · Graphic group lattices and (directed) Topcode-matrix lattices enable us to build up connections between traditional lattices

Graphic Lattices and Matrix Lattices Of Topological Coding · Graphic group lattices and (directed) Topcode-matrix lattices enable us to build up connections between traditional lattices

Introduction - uni-paderborn.de · Introduction Our main motivation for the research leading to the present paper is the classi cation of extremal even unimodular lattices using automorphisms.

Introduction - uni-paderborn.de · Introduction Our main motivation for the research leading to the present paper is the classi cation of extremal even unimodular lattices using automorphisms.

Non-supersymmetric extremal multicenter black holes with ... · Supersymmetric black hole composites ‣ extremal multi-RN solutions are susy ‣ susy (hence extremal) multicenter

Non-supersymmetric extremal multicenter black holes with ... · Supersymmetric black hole composites ‣ extremal multi-RN solutions are susy ‣ susy (hence extremal) multicenter

Isolating a Vertex via Lattices: Polytopes with Totally ...rgurjar/papers/TUM.pdf · Polytopes with Totally Unimodular Faces Rohit Gurjar 1, Thomas Thieraufy2, and Nisheeth K. Vishnoi3

Isolating a Vertex via Lattices: Polytopes with Totally ...rgurjar/papers/TUM.pdf · Polytopes with Totally Unimodular Faces Rohit Gurjar 1, Thomas Thieraufy2, and Nisheeth K. Vishnoi3

Extremal Points

Extremal Points

Exploring Applications of Extremal Optimization

Exploring Applications of Extremal Optimization

The Search for Extremal Graphs

The Search for Extremal Graphs

Unimodular Random Triangulations: Circle Packing …...Unimodular Random Triangulations: Circle Packing and Random Walk Omer Angel, Tom Hutchcroft, Asaf Nachmias, Gourab Ray Abstract

Unimodular Random Triangulations: Circle Packing …...Unimodular Random Triangulations: Circle Packing and Random Walk Omer Angel, Tom Hutchcroft, Asaf Nachmias, Gourab Ray Abstract

Totally unimodular matrices Introduction to integer ...artax.karlin.mff.cuni.cz/~branm1am/download/06_Branda_VRP_2018_PRINT.pdf · Totally unimodular matrices and network ows Minimum

Totally unimodular matrices Introduction to integer ...artax.karlin.mff.cuni.cz/~branm1am/download/06_Branda_VRP_2018_PRINT.pdf · Totally unimodular matrices and network ows Minimum