Self-Dual Codes over� �

and NegacirculantConference Matrices

�

K.T. Arasu and Yu Qing ChenDept. of Mathematics & Statistics

Wright State University, Dayton, OH 45435, USA

T. Aaron GulliverDept. of Electrical & Computer Engineering

University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC, V8W 3P6 Canada

Weilai SongDept. of Mathematics & Statistics

Wright State University, Dayton, OH 45435, USA

Abstract— Previously, self-dual codes ternary have been con-structed from conference matrices. In this paper, we present codesconstructed from negacirculant conference matrices. A necessarycondition for these codes to be self-dual is given, and examplesare given for lengths up to 108. The equivalence with the Plesssymmetry codes is established.

I. INTRODUCTION

A ternary linear � � � � code � over � is a -dimensional

vector subspace of �� . The elements of � are called code-

words and the weight � � � � � of a codeword � is the number

of non-zero coordinates in � . The minimum weight of � is

defined as � � � � � ! # % ' � � � � � + - /� � 2 � 5 7A linear � � � � code � is optimal if � has the highest minimum

weight among all linear � � � � codes (see [2] for current bounds

on the highest minimum weight for small 9 ).

A matrix whose rows generate the code � is called a genera-

tor matrix of � . Two codes � and � ; over � are equivalent if

one can be obtained from the other by a monomial permutation

of the coordinates (that is, by a permutation of the coordinates

followed by multiplication of the coordinates by non-zero field

elements). The dual code � < of � is defined as� < � ' � 2 �@ + � C D � - for all D 2 � 5 7� is self-dual if � � � < . A self-dual [ � � � J L ] ternary code

exists if and only if � M - � ! P R S � . In addition, all weights in

a self-dual ternary code are divisible by 3. A ternary self-dual

code is extremal if � � � � � T V �W L X Y T 7All ternary self-dual codes of length [ L S have been classified

in [3], [7], [8] and [10]. Two families of self-dual codes,

namely the extended quadratic residue codes and the Pless

symmetry codes, are well known (cf. [8]). In these families, the

1This work was supported by NSA and AFOSR. A preliminary versionof this work was presented at the XXVIIth Ohio State-Denison MathematicsConference, June 11–13, 2004, The Ohio State University, Columbus, Ohio.

first few codes are extremal ternary self-dual codes. For larger

lengths, extremal ternary self-dual codes exist for lengths[ S _ � a b � b - and b S , and do not exist for lengths d L � e b � W L - �and all lengths f W S S [11].

A conference matrix g of size h Y Wis an � h Y W � i � h Y W � -

matrix with entries - and j Wwhich satisfiesg C g m � h C p 7

Here p denotes the identity matrix and the superscript qdenotes “transpose”. Many examples of conference matrices

are known and conference matrices were used to construct

self-dual codes in [1].

In [4], the so called negacirculant conference matrices

were studied. A matrix r is called negacirculant when the

following holds: If � s t � 7 7 7 � s w � is the x -th row of M then� z s w � s t � s { � 7 7 7 � s w } { � is the � x Y W � -st row of r . The set

of negacirculant matrices of size h Y Wover a field is an

algebra isomorphic to � � � J � � w � { Y W � . If the negacirculant

conference matrix with first row � s t � 7 7 7 � s w � is denoted by

neg � s t � 7 7 7 � s w � , then the isomorphism is simply

neg � s t � 7 7 7 � s w � � w� � � t s � ��

7 (1)

In [4] it is conjectured that h has to be a prime power if a

negacirculant conference matrix exists.

There is an easy construction of such matrices using relative

difference sets. A relative difference set with parameters� � � � � � � � is a subset � � � of a group � which has the

following properties:

(R1) � � � C � .

(R2) � contains a normal subgroup � of order � .

(R3) + � + � .

(R4) Every element in � � � has exactly � representation s as

a difference � z � ; with elements from � . No element in� � ' - 5 has such a representation.

If � � Wwe have the definition of difference sets in

the usual sense. For obvious reasons, the subgroup � is

called the forbidden subgroup. Note that � meets each coset

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of � at most once. Otherwise, elements in � would have

difference representations with elements from � . Now let � �be another group. If there is an epimorphism � � � � whose

kernel � � of order � � is contained in � then the image of� under this epimorphism is a relative difference set with

parameters � � � � � � � � � � � . This can be easily seen using

the notion of group rings. For this purpose we write the group

multiplicatively. We consider the group ring� � " $ & '( * , . ( � 0 2 . ( 3 � 4 where addition is defined componentwise, and multiplication

is ' ( . ( 0 � � ' ( 8 ( 0 � $ ' ( ' ; .;

8 ( ; < =� 0 @

If A is a subset of � , we identify A with B ( * D 0 3 G � " ,i.e. the coefficient of elements in A is

I, otherwise it is J .

Moreover, if A $ B ( * , . ( 0 , we define A M O P 2 $B ( * , . ( 0 O . The set � is a relative � � � � � -difference set

in � relative to � if and only if� � � M V W P $ � X � � Y � � (2)

holds in� � " . Note that any epimorphism [ 2 � � � � _

can be extended to an epimorphism of the group ring� � " �� � � _ " . Let b _ b $ � � . We denote the image of � 3 � � "

under this projection by � g and call this group ring element

a difference list. We have

� g � � M V W Pg $ � X � � � � g Y � b � l _ b � g @If _ n � then b � l _ b $ � � and � g has coefficients just Jand

I. Therefore, � g is again a relative difference set.

Lemma 1. Let � be a � � � � � -difference set in � relativeto � . If _ is a normal subgroup of � contained in � then� g is a � � � b _ b � � b _ b � -difference set in � � _ relative to� � _ .

In this paper we are interested in the case � $ � X I.

Then the set � meets every coset of � in one element

with one exception, the exceptional coset is disjoint from � .

The following construction yields cyclic difference sets with

parameters u X I u Y I u I � (3)

where u is a prime power:

Theorem 2. Let x y z be the finite field with u y elements. Theset & | 3 x y z � & J 4 2 | X | z $ I 4is a cyclic relative difference set with parameters � � in themultiplicative group of x y z .

Difference sets with these parameters are called affine. Ifu is odd, projection onto the cyclic group of order � u X I �yields a circulant relative u X I � u u Y I � � � � -difference

set � in� y M z � W P . These relative difference sets give rise to

negacirculant conference matrices. One may think of the group

ring� � y M z � W P " as the quotient

� | " � | y M z � W P Y I � of the

polynomial ring� | " modulo | y M z � W P Y I

. Then the image

of � in� | " � | z � W X I � (interpreted as a matrix, see (1)), is

the desired negacirculant conference matrix. It is also possible

to reverse this construction: Using Theorem 2, we obtain the

following family of negacirculant conference matrices:

Theorem 3. Negacirculant conference matrices of size u X I equivalently cyclic u X I � u u Y I � � � � -relative differencesets � exist if u is a prime power.

In [4], the first systematic investigation was carried out to

find necessary conditions on these matrices. It was shown up

to u $ � � � that u has to be an odd prime power.

II. SELF-DUAL CODES FROM NEGACIRCULANTCONFERENCE MATRICES

We consider ternary codes of the form � � " where � is a negacirculant conference matrix of size u XI

. Note that these codes belong to the class of quasi-twistedcodes [5]. To construct ternary self-dual codes, since u must

be odd, u X Imust be a multiple of 6. Using the construction

given in the previous section, the first rows of the negacirculant

conference matrices are given below.u First Row� � I � J I II I � I I � I � J � � � � II � I I � I � � � I � J I I I � I I I I� � I I � I I I I I I � � I J I I � � I � I � I I I� � � I � � � � � I I I � I I � I J � � � I I I � � I � I � I I� I I � � � � � I � � � � I � I � � I � � I I J � I I � � � I I I I I � I � � � I � I �� � I I � I � � � � I � I � I � � � I I � � � I I � J � � I I � I I � � I � � � � � � �I � I I I I� � � I � I � I I � � � I � � � I � I � � I � � I � I I � J I I � � � � I I I � � � � �I � � � I � � I I I I IThe minimum distances of the corresponding codes areu � u �

� � � � I �I I � � I � II � I � � � � �� � I � � � � � �One may notice that the parameters of these codes coincides

exactly with those of the Pless symmetry codes [9], which are

ternary self-dual codes constructed using Paley matrices. We

now show that these codes are in fact equivalent.Let u be an odd prime power and x z the finite fields with u

elements. Paley matrices of order u X Iare defined as follows.

Definition 4. Let � $ & � � $ | � � � � � W $ | W � W � � � � � z $ | z � z � 4 be a set of u X Ipairwise linearly

independent � -dimensional vectors over x z . We define a Paleymatrix � � associated with � as

� � ¡ ¢£¤ ¥ ¦ § ¨ © ¦ « ¬ ­ « ¬ ® ® ¥ ¦ § ¨ © ¦ « ¬ ­ « ° ® ® ² ² ² ¥ ¦ § ¨ © ¦ « ¬ ­ « ³ ® ®¥ ¦ § ¨ © ¦ « ° ­ « ¬ ® ® ¥ ¦ § ¨ © ¦ « ° ­ « ° ® ® ² ² ² ¥ ¦ § ¨ © ¦ « ° ­ « ³ ® ®² ² ² ² ² ²¥ ¦ § ¨ © ¦ « ³ ­ « ¬ ® ® ¥ ¦ § ¨ © ¦ « ³ ­ « ° ® ® ² ² ² ¥ ¦ § ¨ © ¦ « ³ ­ « ³ ® ®´ µ¶ ­

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where � � � � � � � � � � � � � � � � � � � � � � �

and ! is the quadratic character of " # with the stipulationthat ! � & � � & .

Proposition 5. Let ( ) � + � ) � , - - - � # 1 ) 3 and ( , �+ � 5 ) � 5 , - - - � 5 # 1 ) 3 be two sets of 7 8 9pairwise linearly

independent vectors over " # . There exists a signed permutationmatrix : such that ; = > � : ; = @ : A ) .

Proof: Let B be a C -dimensional vector space over " # .

Then B contains 7 8 9distinct lines passing through the origin� & & � and they are the points of the projective line G ) " # . Both

the set ( ) and ( , contain one point from each line in G ) " # .

Therefore there must be a permutation J on G ) " # and 7 8 9non-zero elements K ) K , - - - K # 1 ) in " # such that � � K � 5 M N Ofor all P � 9 C - - - 7 8 9

and ; = > � : ; = @ : A ), where

: � STTTU! � K ) � ! � K , �

. . . ! � K # 1 ) �V W WWX

Y the permutation matrix of J ZThe proposition shows that all Paley matrices are equivalent.

We now look at the following two Paley matrices.First let[ \ ] _ ` b c d e c ` g \ c b e c ` g � c b e c i i i c ` g k c b e � _ g \ c g � c i i i c g k � ] � k � �

Then

m n > ] opppq d b b i i i br ` � b e d r ` g \ � g � e i i i r ` g \ � g k er ` � b e r ` g � � g \ e d i i i r ` g � � g k ei i i i i ir ` � b e r ` g k � g \ e r ` g k � g � e i i i dw xxxy �

For the second one, let " # @ be the quadratic extension of " # .

Let be an element of " # @ such that {| " # but , | " # . Then" # @ � + � 8 � � � � | " # 3 ZWith this representation, the Galois action is given by� � 8 � � # � � � � and the trace map from " # @ to " # is given by� � # @ # � � 8 � � � C � ZFor any two elements � � � ) 8 � , and � � � ) 8 � , in " # @ ,

we define� � � � � � � � � � � � � ) � ,� ) � , � � � ) � , � � , � ) � � � # @ # � � � #� C �Let � be a primitive element of " # @ . Then ( , �+ 9 � � , - - - � # 3 contains 7 8 9

elements of " # @ that are

pairwise linearly independent over " # . The Paley matrix ism n @ ]opppq r ` � � � ` � � c � � e e r ` � � � ` � � c � \ e e i i i r ` � � � ` � � c � k e er ` � � � ` � \ c � � e e r ` � � � ` � \ c � \ e e i i i r ` � � � ` � \ c � k e e...

...r ` � � � ` � k c � � e e r ` � � � ` � k c � \ e e i i i r ` � � � ` � k c � k e ew xxxy

] oppppq r ` � � k @ � k ` � � � � � �� � ! e e i i i r ` � � k @ � k ` � � � � � �� � ! e er ` � � k @ � k ` �>

�>

� �� � ! e e i i i r ` � � k @ � k ` �>

� � � �� � ! e e......r ` � � k @ � k ` � � � � � �� � ! e e i i i r ` � � k @ � k ` � � � � � �� � ! e e

w xxxxy �

Let " be the matrix with ! � � A � N # 1 ) O � � C , � � ,! � � A ) N # 1 ) O � � C , � � Z Z Z ! � � A # N # 1 ) O � � C , � � on the diagonal.Then

m ' ] opppq r ` � � k @ � k ` � � � � ) e e i i i r ` � � k @ � k ` � � � k ) e er ` � � k @ � k ` � \ � � ) e e i i i r ` � � k @ � k ` � \ � k ) e e......r ` � � k @ � k ` � k � � ) e e i i i r ` � � k @ � k ` � k � k ) e e

w xxxySince � is primitive, one has ! � � # 1 ) � � � 9

and! � � � # @ # � � N # 1 ) O A � � � � � ! � � � # @ # � � A � � � ZIf we define

� A ) � STTTTTU9 9

. . . 9� 9V WWWWWX

then ; " � #�+ � � ! � � � # @ # � � A + � � � A + ZTherefore, ; " is negacirculant. Furthermore, � - � + � / � � � � , - - - � � # 3 0 � , � - { , � , N # 1 ) O - ZUnder the isomorphism2 � , � - � , � - { , � , N # 1 ) O - 2 � � � � � the set + ! � � � # @ # � � A + � � � A + � � 9 C - - - 7 3corresponds to+ � N # 1 ) O N ) A 6 N 8 9 � @ : � N <

¢ = > O O O , � A + � � 9 C - - - 7 3which is the translation of+ � N # 1 ) O N ) A 6 N 8 9 � @ : � N <

¢ = > O O O , � A + � � 9 C - - - 7 3by A )

. Since! � � � # @ # � � N # 1 ) O N ) A 6 N 8 9 � @ : � N <¢ = > O O O , � A + � �� ! � � � # @ # � � A + � � � , � 9

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the set

� � � � � � � � � � � � � � � � � �

�� � � � � � � � � � � � � � � � � � � � � �

is the � � � � � � � � � � # � $ % � $ -relative difference set obtained

from the affine � � � � � � � # � � � $ -relative difference set

� � + , . �� � / 0 � � � � � � + $ � � �in . �� � by taking the quotient . �� � % . �� . Hence the Paley matrix3 4 � is equivalent to the negacirculant matrix obtained from

the relative difference set. Since 3 4 6is equivalent to 3 4 � ,

the Paley matrix 3 4 6is also equivalent to the negacirculant

matrix from the � � � � � � � � � � # � $ % � $ -relative difference set.

III. WEIGHT CALCULATION

One of the advantage of constructing codes from combi-

natorial designs is that weights of some codewords can be

calculated by using the structure of the design. In this section,

we discuss how to calculate weights of codewords combining

less than four rows of the generator matrices of the codes

obtained from negacirculant matrices. From the isomorphism

between the matrix group� : � and cyclic group

� � � given

in the previous section, we can see that the ternary code

constructed from a relative difference set ; in the cyclic group� � � is the< = ? � � � A % � � � � � � $ submodule generated by

� � ;in < = ? � � � A % � � � � � � $ � < = ? � � � A % � � � � � � $ DThere is a natural surjective ring homomorphism< ? � � � A E < = ? � � � A % � � � � � � $ �and the kernel of this homomorphism is � F � � � � � � $ D Hence

every module over< = ? � � � A % � � � � � � $ can be regarded

as a< ? � � � A module. A rather obvious consequence of this

viewpoint is the following theorem.

Theorem 6. Every codeword in the ternary code generatedby ; corresponds to a unique element of the form H ; in< ? � � � A , where H is a subset of

� � � which contains no cosetsof

� � � � � � . The weight of H ; equals / H / plus the numberof elements of the form K L M L � � � � appearing in H ; with

K NO M � Q S T F $ , where K and M are non-negative integers, andL , � � � .For example, in case � � Z , we have ; � � � � " � # � $ , which corresponds to the first row of : " in Section

2, while � # ; corresponds to the negative of the first row. The

element

� � � $ ; � � � � $ � � � � " � # � $ $� � � � � � � " � � # � & � $ � 'corresponds to the sum of the first two rows of : " and can

be written as� ] � ^ � � # $ � � � & $ � � � � $ $ � ] � = � ' $ � � ] � � ^ $ � � � " ] � � � $ D

By Theorem 5, its weight is � _ � b . Similarly, � � � & $ ;corresponds to the difference of the first two rows and its

weight is also � _ � b . From the definition of relative

difference set, we have

Corollary 7. If H is a two-element subset of� � � and not a

coset of� � � � � � , then the weight of H ; is F � � � $ % � .

Corollary 8. If H is a three-element subset of� � � containing

no cosets of� � � � � � , then the weight of H ; isF � � � �

� K $ �where ] , K , F is an integer.

Proof: Let L , f , K and M be the number of elements of the

form ] g F g � � � �, g � g � � � �

, ] g � g � � � �and g g � � � �

,

respectively, appearing in H ; . Clearly, K M � F and L f �� # � . From

; ; � � � � # �� � � � � # � � � � � � $ �

one has

� H ; $ � H ; $ � � H H � � � � # �� � � � � # � � � � � � $ $ D

Comparing the coefficients of�

and � � � �, one findso L f _ K � F �_ f � M � F � � # � $ (4)

From (4), one has

f K � F � � � # F� K $ DHence the weight of H ; is

F f K � F � � � �� K $ D

Length 96 was the computational limit of the minimum

distance calculations. Determining that

r� � _ for the [96,48]

code took 53 days, and 12 days to find that

r� � �

for the

[84,42] code. Determining that

rs � _ for the [108,54] code

took only 2 days.

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[2] A.E. BROUWER, Bounds on the size of linear codes, in Handbook ofCoding Theory, eds. V. Pless et al., Elsevier, Amsterdam, 1998.

[3] J.H. CONWAY, V. PLESS AND N.J.A. SLOANE, Self-dual codes over2 / 4 6 8 and2 / 4 : 8 of length not exceeding ; = , IEEE Trans. Inform.

Theory, 25 (1979), pp. 312–322.[4] P. DELSARTE, J. GOETHALS, AND J. SEIDEL, Orthogonal matrices with

zero diagonal, II, Canad. J. Math., 23 (1971), pp. 816–832.[5] R. HILL AND P.P. GREENOUGH, Optimal quasi-twisted codes, Proc. Int.

Workshop Algebraic and Comb. Coding Theory, pp. 92-97, June 1992.[6] D. JUNGNICKEL AND A. POTT, Perfect and almost perfect sequences,

Discrete Applied Math., 95 (1999), pp. 331–359.[7] J.S. LEON, V. PLESS AND N.J.A. SLOANE, On ternary self-dual codes

of length > : , IEEE Trans. Inform. Theory, 27 (1981), pp. 76–180.[8] C.L. MALLOWS, V. PLESS AND N.J.A. SLOANE, Self-dual codes over2 / 4 6 8 , SIAM. J. Appl. Math. 31 (1976), pp. 649–666.[9] V. PLESS, Symmetry codes over

2 / 4 6 8 and new five-designs, J.Combin. Theory Ser. A, 12 (1972), pp. 119–142.

[10] V. PLESS, N.J.A. SLOANE AND H.N. WARD, Ternary codes of mini-mum weight = and the classification of self-dual codes of length > A ,IEEE Trans. Inform. Theory, 26 (1980), pp. 305–316.

[11] E.M. RAINS AND N.J.A. SLOANE, Self-dual codes, in Handbook ofCoding Theory, eds. V. Pless et al., Elsevier, Amsterdam, 1998.

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