A Generalized Construction of Extended Goppa Codes

5296 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 9, SEPTEMBER 2014

A Generalized Construction of ExtendedGoppa Codes

Mubarak Jibril, Sergey V. Bezzateev, Martin Tomlinson, Markus Grassl,and Mohammed Zaki Ahmed

Abstract— We present a generalized construction of extendedlength Goppa codes. Using this construction, we obtain 71 newcodes in finite field Fq for q = 4, 7, 8, 9 with better minimumdistance than the previously known codes with the same lengthand dimension.

Index Terms— Goppa, BCH, Reed Solomon, error correction.

I. INTRODUCTION

GOPPA introduced a class of linear codes in [1] and[2], commonly referred to as Goppa codes or �(L, G)

codes. �(L, G) codes have good properties and some of thesecodes have the best known minimum distance of any knowncodes with the same length and rate. Several methods ofextending Goppa codes by improving both their length andminimum distance have been presented in literature [3]–[5].In [5] the authors present a construction of extended Goppacodes which is generalisation of the method in [4] for binaryGoppa codes and produce many improved codes in F7,F8 and F9. In [5] the authors use Goppa polynomials whoseunique factors have at most one root. In this paper we suggestthe construction of extended Goppa codes defined with poly-nomials whose unique factors have more than one root. Thiscan be seen as a generalisation of the method in [5]. As a resultwe produce codes in F4, F7, F8 and F9 that have better mini-mum distances than the codes in [6] with the same length anddimension.

Section II gives a brief background on Goppa codes anda description relevant to the construction of extended Goppacodes. Section III gives a generalisation of the method in [5]and establishes the parameters of nonbinary codes obtainedtherefrom. Section IV shows that codes with better dimensionsthan the designed dimension can be obtained by choosingGoppa codes which are subcodes of BCH codes. In Section VI

Manuscript received February 17, 2014; revised May 17, 2014; acceptedJune 8, 2014. Date of publication June 13, 2014; date of current versionAugust 14, 2014.

M. Jibril is with the Nigerian Communications Satellite Ltd., Abuja 932001,Nigeria (e-mail: [email protected]).

S. V. Bezzateev is with the Department of Technologies of InformationSecurity, Saint Petersburg State University of Aerospace Instrumentation,St. Petersburg 190000, Russia (e-mail: [email protected]).

M. Tomlinson and M. Z. Ahmed are with the School of Computing andMathematics, University of Plymouth, Plymouth PL4 8AA U.K. (e-mail:[email protected]; [email protected]).

M. Grassl is with the Centre for Quantum Technologies, National Universityof Singapore, Singapore 117543 (e-mail: [email protected]).

Communicated by N. Kashyap, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2014.2330814

we exploit the nested structure of these extended Goppa codesby using Construction X [7] to produce further improvements.Finally, Section VII gives a summary of the new codes foundusing the construction method.

II. PRELIMINARIES

A. Goppa Codes

A �(L, G) code is defined by a set L ⊆ Fqm and apolynomial G(x) with coefficients from Fqm , where Fqm is afinite extension of the field Fq . The set L = {α0, α1, . . . , αn−1}with cardinality n contains different elements of Fqm that arenot roots of the Goppa polynomial G(x). Formally, L ⊆{β ∈ Fqm : G(β) �= 0}. As is usually done, we choose theset L of the Goppa code such that |L| is maximal. A word(a0, a1, . . . an−1) with elements from Fq is a codeword of aGoppa code defined by the set L and the polynomial G(x) ifit satisfies

n−1∑

i=0

ai

x − αi≡ 0 mod G(x). (1)

If r is the degree of the polynomial G(x) ∈ Fqm [x] theparameters of the Goppa code are:

length: n = |L|,redundancy: n − k ≤ mr,

distance: d ≥ r + 1.

A parity check matrix of a Goppa code is given by

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

G(α0)

1

G(α1)· · · 1

G(αn−1)α0

G(α0)

α1

G(α1)· · · αn−1

G(αn−1)α2

0

G(α0)

α21

G(α1)· · · α2

n−1

G(αn−1)...

.... . .

...

αr−10

G(α0)

αr−11

G(α1)· · · αr−1

n−1

G(αn−1)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (2)

As with the representation of Goppa codes in [1]–[3] and [9]the entries of their parity check matrices are expressed herewith symbols from the finite field Fqm , it is assumed thatparity check matrices whose entries are symbols of the sub-field Fq are easily obtained in a manner described in theAppendix.

0018-9448 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

JIBRIL et al.: GENERALIZED CONSTRUCTION OF EXTENDED GOPPA CODES 5297

Theorem 1: Suppose the polynomial G(x) is factorisableinto � factors such that,

G(x) =�∏

μ=1

Gμ(x),

and Gμ(x), μ = 1, . . . , � have no common factors i.e.

GC D(Gi (x), G j (x)) = 1, ∀i �= j,

where GCD denotes the greatest common divisor. Let theset L = {α0, . . . , αn−1} be defined such that G(β) �= 0,∀β ∈ L and a codeword a ∈ �(L, G), then a = (a0, . . . , an−1)satisfies,

a · H T�(L ,G) = 0, H�(L ,G) =

⎡⎢⎢⎢⎢⎣

H�(L ,G1)

H�(L ,G2)

...

H�(L ,G�)

⎤⎥⎥⎥⎥⎦

(3)

where H�(L ,Gμ) is the parity check matrix of the Goppa codedefined with polynomial Gμ(x) and set L.

Proof: Consider the set of congruences,

F(x) ≡ 0 mod Gμ(x) μ = 1, . . . , � (4)

where F(x) is in the ring Fq[x]. By definition F(x) ≡∑n−1i=0

aix−αi

mod G(x) for a ∈ �(L, Gμ) for all μ. Eachcongruence relation with respect to the polynomial Gμ(x)defines a Goppa code �(L, Gμ). Equation (4) implies thatF(x)|Gμ(x) for all μ. Since GC D

(Gi (x), G j (x)) = 1,

∀i �= j , clearly F(x)|(∏�

μ=1 Gμ(x))

. As such,

F(x) ≡ 0 mod G(x) and a ∈ �(L, G). (5)

Thus if (4) holds, (5) also holds. As such a ∈ �(L, G) satisfiesthe parity check matrix in (3).Goppa codes can be extended by adding to their respectiveparity check matrices a row of length |L| + 1 containing all1 entries.

Theorem 2 (From [5, Th. 3]): A linear code defined in thefield Fq with the parity check matrix,

H̃e =[

11 . . . 1 1H 0

]. (6)

where H is the parity check matrix of a Goppa code definedwith a polynomial G(x) having coefficients in Fqm and the setL ⊆ {β ∈ Fqm : G(β) �= 0} has parameters,

length: n = |L| + 1,

redundancy: n − k ≤ mr + 1,


An interesting relation between the parity check matrix of aGoppa code (defined with a polynomial whose factors aremonic polynomials with degree equal to 1) and the Cauchymatrix was presented in [4] and more explicitly in [8].

Theorem 3 (From [8, Appendix]): Let γ in Fqm . A�(L, G)defined by a polynomial G(x) = (x − γ )r , satisfies the parityequations

n−1∑

i=0

ci

(γ − αi ) j= 0. j = 1, . . . , r

The parity check matrix of the code can be expressed as

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

(γ − α0)

1

(γ − α1)· · · 1

(γ − αn−1)1

(γ − α0)2

1

(γ − α1)2 · · · 1

(γ − αn−1)2

......

. . ....

1

(γ − α0)r

1

(γ − α1)r· · · 1

(γ − αn−1)r

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (7)

This code with symbols in Fq and defining set L = Fqm \γ ={α0, . . . , αn−1} has parameters

length: n = |L|,redundancy: n − k ≤ mr,


III. CODE CONSTRUCTION

The construction presented below is a generalisation ofConstruction P in both [4] and [5]. Consider the Goppapolynomial defined with coefficients from the field Fqm ,

G(x) =�∏

μ=1

Grμμ (x) (8)

where Gμ(x) is a distinct monic irreducible polynomial ofdegree dμ and rμ is a positive integer. Polynomials Gμ(x) areirreducible in the ring Fqm [x]. In addition we set the restrictionrμ = 1 for dμ > 1. The Goppa polynomial G(x) has degreer = ∑�

μ=1 dμrμ. The set L for the polynomial G(x) is givenby L = {β ∈ Fqm : G(β) �= 0}. The codes CP are defined withparity check matrix

HP =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

11 . . . 1 1 0 · · · 0H1 0 HI1 · · · 0...

......

. . ....

H� 0 0 · · · HI�

0 0 HK1 · · · 0...

......

. . ....

0 0 0 · · · HK�

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (9)

The first |L|+1 columns of HP contain the parity check matrixof the extended Goppa code defined by G(x),

H̃e =

⎡⎢⎢⎢⎣

11 . . . 1 1H1 0...

...H� 0

⎤⎥⎥⎥⎦. (10)

where Hμ ≡ H�(L ,G

rμμ )

, μ = 1, . . . , �. This representationfollows Theorem 1. Additionally Hμ is in the form of (7)when dμ = 1 and in the form of (2) when dμ > 1. HIμ aredμrμ × nμ matrices of the form,

HIμ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · 0 0 · · · 0...

. . ....

.... . .

...0 · · · 0 0 · · · 0

Ha · · · 0 0 · · · 0...

. . ....

.... . .

...0 · · · Ha 0 · · · 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.


whose dμ(rμ − 1) topmost rows and (nμ − mdμ) rightmostcolumns1 have zero entries. The submatrix Ha is 1 × m rowmatrix of the form

Ha = [1 α α2 · · · αm−1

]

where α is the primitive element of Fqm . The matrix HKμ

is a parity check matrix of code defined in the subfield Fq

with length nμ, dimension mdμ and minimum distance dμ.The columns of HKμ are arranged such that the leftmost mdμ

columns are linearly independent. Clearly the code CP haslength and redundancy,

length: n = |L| + 1 +�∑

μ=1

nμ,

redundancy: n − k ≤ mr + 1 + ∑�μ=1

(nμ − mdμ

). (11)

For the case where dμ = 1 for all μ, this construction isequivalent to the construction in [5, Sec. III]. To obtain a lowerbound on the minimum distance of the codes CP, we can followthe logic and presentation of the [4, Proof of Theorem 7].

Theorem 4: The minimum distance of the code CP is lowerbounded by d ≥ r + 2.

Proof: Let c = (c0, c1, . . . , c�) be a codeword of CP, where

c0 = (a0, a1, . . . , a|L|−1, a|L|)

ai ∈ Fq and cμ ∈ Fnμq for 1 ≤ μ ≤ �.

If at least one of these vectors cμ for 1 ≤ μ ≤ � is non-zeroin the leftmost mdμ coordinates, then c0 must be non-zero aswell since the leftmost mdμ columns of the submatrices HIμ

are linearly independent over Fq . Additionally the leftmostmdμ columns of the corresponding parity check matrix HKμ

are linearly independent2 and cμ satisfies HKμ . As a resultwhen cμ is non-zero, cμ has weight at least dμ.

Therefore, assume that c0 is non-zero. Furthermore, let UZ

and UN be the sets of integers μ such that cμ, μ ≥ 1 is zeroor non-zero, respectively. For μ ∈ UN, by definition cμ �= 0,and cμ has weight at least dμ. Hence the weight of c is lowerbounded by wgt(c0) + ∑

μ∈UNdμ.

In order to obtain a bound on the weight of c0 first notethat cμ �= 0 implies that the parity check given by the lastdμ rows of Hμ in HP do not hold for c0, but the other paritycheck equations are fulfilled. Hence for cμ �= 0 (μ ∈ UN) wecan remove the last dμ rows of Hμ in HP. The resulting paritycheck matrix is that of an extended Goppa code and one thatc0 satisfies (since cμ = 0 and μ ∈ UZ). The extended Goppacode is defined by the Goppa polynomial,

G̃(x) =∏

μ∈UN

Grμ−1μ (x)

∏

μ∈UZ

Grμμ (x)

The degree of G̃(x) is r −∑μ∈UN

dμ, and hence by Theorem 2wgt(c0) ≥ r − ∑

μ∈UNdμ + 2. In summary we get

wgt(c) ≥ wgt(c0) + ∑μ∈UN

dμ ≥ r + 2.

1The code defined by parity check matrix HKμ has redundancy (nμ−mdμ).2The leftmost mdμ coordinates of cμ are information symbols of the code

defined by HKμ .

IV. CP AS EXTENDED BCH CODES

In [5] it is shown that the code Cp can be defined as extendedBCH code. In this section we follow the logic in [5] to showthat the code CP defined with a Goppa polynomial G(x) and aset L is an extension of a Goppa code �(L, F) defined withthe polynomial F(x), a factor of G(x) and the set L. As aresult CP has the same dimension as the Goppa code �(L, F).Secondly we show the choice of polynomial F(x) such thatthe codes CP are extensions of BCH codes.

Theorem 5: Let F(x) = ∏�μ=1 G

rμ−1μ (x) where

G(x) = F(x)

�∏

μ=1

Gμ(x).

Then the code CP defined with the Goppa polynomialin (8) is an extension of the Goppa code �(L, F) withL = {β ∈ Fqm : G(β) �= 0}. Furthermore the dimension ofCP is equal to the dimension of �(L, F).

Proof: Let c0 = (a0, a1, . . . , a|L|−1) be a codeword ofthe Goppa code �(L, F) with L = {β ∈ Fqm : G(β) �= 0}and parity check matrix of the code is H0. We define themap πm that maps each element of Fqm to a unique vectorin F

mq ,

πm : Fqm → Fmq

πm(β) = (b0, b1, . . . , bm−1) β ∈ Fqm , bi ∈ Fq .

We use (1, α, . . . , αm−1) as a basis of the vector spaceF

mq such that β = b0 + b1α + · · · + bm−1α

m−1. Secondly

we use σdμ

mdμ,nμto represent an encoding map from an

mdμ-dimensional message space in Fq to an nμ-dimensionalcode space,

σdμ

mdμ,nμ: F

mdμq → F

nμq

with mdμ ≤ nμ and the requirement that the nμ-dimensionalcode has minimum distance dμ. For brevity we replace the

notation σdμ

mdμ,nμwith σμ. We can represent c ∈ CP in the

form

c = (a0, . . . , a|L|−1, a|L|, σ1(c1), σ2(c2), . . . , σ�(c�)),

where a j ∈ Fq and cμ ∈ Fmdμq . Each cμ is,

cμ = (πm(cμ,0), . . . , πm(cμ,dμ−1)

)

where cμ, j ∈ Fqm .Let hμ be a dμ × |L| matrix such that

hμ =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

Gμ(α0)· · · 1

Gμ(αn−1)α0

Gμ(α0)· · · αn−1

Gμ(αn−1)...

. . ....

αdμ−10

Gμ(α0)· · · α

dμ−1n−1

Gμ(αn−1)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

when dμ > 1 and

hμ =[

1

(γμ − α0)rμ

1

(γμ − α1)rμ· · · 1

(γμ − αn−1)rμ

].


when dμ = 1 and Gμ(x) = (x − γμ)rμ . The parity checkmatrix of code CP defined with G(x) in (8) can be representedas,

HP =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 · · · 1 1 0 · · · 0h0 0 0 · · · 0h1 0 HI1 · · · 0...

......

. . ....

h� 0 0 · · · HI�

0 0 HK1 · · · 0...

......

. . ....

0 0 0 · · · HK�

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

Note that this matrix is expressed in a manner described in theAppendix and has entries from the subfield Fq . The submatrixHIμ is a dμ × nμ matrix,

HIμ =⎡

⎢⎣Ha · · · 0 0 · · · 0...

. . ....

.... . .

...0 · · · Ha 0 · · · 0

⎤

⎥⎦

and

Ha = [1 α α2 · · · αm−1

].

From the first row of the parity check matrix in (12) we have,

a|L| = −|L|−1∑

i=0

ai

and for rows that contain the submatrix hμ we have

cμ, j = −|L|−1∑i=0

αji

Gμ(αi )for j = 0, . . . , dμ − 1.

when dμ > 1 and

cμ, j = −|L|−1∑i=0

1

(γμ − αi )rμfor j = 0

when dμ = 1. Thus CP is an extension of the Goppa code�(L, F). Since there exists a one to one mapping betweenthe codewords of �(L, F) and codewords of CP, the codeshave the same dimension.

Remark 1: Observe that Theorem 5 improves the lowerbound on the dimension of the codes CP in (11) to k = k�(L ,F),where k�(L ,F) is the dimension of the Goppa code �(L, F).BCH codes can be defined as Goppa codes with an appropriatechoice of a Goppa polynomial.

Theorem 6 (See [9, Ch. 12, Sec. 3, Problem (6)]): A Goppacode �(L, G) defined with the polynomial G(x) = xr , andthe set L = Fqm \ {0} is equivalent to a BCH code defined inFq with length n = qm − 1.

Proof: See [5, Proof of Th. 6].The true dimensions of BCH codes are well known and aremuch greater than the lower bound k ≥ |L| − mr when thecodes are defined with the Goppa polynomial xr and set L [9].We set the polynomial F(x) = xr1−1 such that �(L, F) is ashortened3 BCH code. As such the code CP is defined with the

3�(L , F) is a shortened BCH code if L �= Fqm \ {0}.

TABLE I

NEW CODES CP OVER F7 WITH THE GOPPA POLYNOMIALS

OVER F49 WHERE α IS A ROOT OF x2 − x + 3

Goppa polynomial

G(x) = xr1

�∏

μ=2

Gμ(x)

Theorem 7 states this explicitly.Theorem 7 The code CP defined with an instance of the

Goppa polynomial in (8) given by,

G(x) = xr1

�∏

μ=2

Gμ(x) (13)

is the extension of a BCH code defined with F(x) = xr1−1

and L = {β ∈ Fqm : G(β) �= 0} with parameters,

length: n = |L| + 1 + ∑�μ=1 nμ,

dimension: k = kBCH,

distance: d ≥ r + 2,

where kBCH is the true dimension of the BCH code.Proof: The statement on the minimum distance and

dimension of the code CP follow from Theorem 4 andTheorem 5, respectively.

V. EXAMPLE

We use as an illustration of the construction a polynomialG(x) = x2(x + 1)(x2 + α2x + α) with coefficients from F16to define an extended Goppa code in F4. The finite field F16is defined with the primitive polynomial s4 + s + 1 and hasα as a primitive element. The set L corresponding to G(x) isthen given by

L = F16 \ {0, 1}, |L| = 14.

Let G(x) = G1(x)G2(x)G3(x) such that G1(x) = x2,G2(x) = x + 1 and G3(x) = x2 +α2 +α. From (6) the paritymatrix H̃ of the extended Goppa code over F16 is given at topof the next page.CP is defined by the parity check matrix HP over F4, given by

HP

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0ω 1 1 0 ω ω ω ω 0 ω 1 ω ω 0 0 0 0 0 0 0 0 0 0 0ω ω ω ω 0 ω 1 ω ω 0 1 ω 1 1 0 0 0 0 0 0 0 0 0 01 0 ω ω ω ω 0 ω 1 ω ω 0 1 ω 0 1 0 0 0 0 0 0 0 0ω ω ω ω 0 ω 1 ω ω 0 1 ω 1 1 0 0 1 0 0 0 0 0 0 00 ω 0 ω ω ω 0 1 ω ω ω 1 ω 1 0 0 0 1 0 0 0 0 0 0ω ω 1 ω 0 1 ω ω 1 0 ω 1 ω ω 0 0 0 0 1 0 0 0 0 0ω ω ω ω 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0ω ω ω 0 ω ω 0 1 ω ω 1 0 ω ω 0 0 0 0 0 0 1 0 0 0ω ω 1 0 0 ω ω 0 ω ω ω ω 1 ω 0 0 0 0 0 0 0 1 0 01 0 0 1 1 ω 1 ω ω 1 1 ω 0 ω 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,


H̃ =

⎡

⎢⎢⎢⎢⎢⎢⎣

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1α14 α13 α12 α11 α10 α9 α8 α7 α6 α5 α4 α3 α2 α 0α13 α11 α9 α7 α5 α3 α α14 α12 α10 α8 α6 α4 α2 0α11 α7 α α14 α5 α2 α6 α13 α8 α10 α3 α4 α9 α12 0α3 α7 α3 α5 α6 α13 1 α4 α12 α12 α4 1 α13 α6 0α2 α5 1 α α α7 α8 α11 α3 α2 α8 α3 1 α7 0

⎤

⎥⎥⎥⎥⎥⎥⎦.

TABLE II


OVER F64 WHERE α IS A ROOT OF x6 + x4 + x3 + x + 1

TABLE III


OVER F81 WHERE α IS A ROOT OF x4 − x3 − 1

where ω is a primitive element of F4. Observe that bydefinition HK1 and HK2 are parity check matrices of [2, 2, 1]4codes with no redundancy and are not represented in HP. HK3,represented by the last row of HP, is the parity check matrixof a [5, 4, 2]4 code defined in F4. Since r = deg G(x) = 5,m = 2 and |L| = 14, the code has parameters [24, 12, 7]4.

TABLE IV

NEW CODES VIA CONSTRUCTION X APPLIED TO CODES CR

OVER F4 WITH THE GOPPA POLYNOMIALS OVER F4

WHERE α IS A ROOT OF x4 + x + 1

TABLE V



WHERE α IS A ROOT OF x2 − x + 3

The minimum weight of the code was confirmed by directcomputation using Magma [10]. Observe that the code CP isan extension of the shortened BCH code [14, 12, 2]4 definedwith Goppa polynomial x and the set L.

VI. NESTED STRUCTURE FROM CODES CP

Construction X [7] is a well known method of extendingnested codes.

Theorem 8 (Construction X [7]): If a linear code C1 withparameters [n1, k1, d1]q has a subcode C2 with parameters[n2, k2, d2]q , then C1 is extendable to a code with para-meters [n1 + n, k1, min{d1 + δ, d2}]q using an auxiliarycode [n, k1 − k2, δ]q .Consider the code CR defined with the Goppa polynomial

G(x) = xr1

�∏

μ=2

Gμ(x).

and the parity check matrix

HR =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

11 . . . 1 1 0 · · · 0H1 0 0 · · · 0H2 0 HI2 · · · 0...

......

. . ....

H� 0 0 · · · HI�

0 0 HK2 · · · 0...

......

. . ....

0 0 0 · · · HK�

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

where H1 is the parity check matrix of the Goppa code�(L, xr1 ) and submatrices HIμ and HKμ are as defined in


TABLE VI



WHERE α IS A ROOT OF x6 + x4 + x3 + x + 1

Section III. Thus if CR and CP are defined with the sameGoppa polynomial G(x) then HR corresponds to HP withthe submatrices HI1 and HK1 removed. The codes CR haveparameters,

length: n = |L| + 1 + ∑�μ=2 nμ,

redundancy: n − k ≤ mr + 1 + ∑�μ=2

(nμ − mdμ

),


Suppose CRu is defined with Gu(x) = xa ∏�μ=2 Gμ(x) and CRv

is defined with Gv (x) = xb ∏�μ=2 Gμ(x) with a < b, then

CRv ⊂ CRu

holds4 provided that the submatrices HKμ , μ = 1, . . . � in HRu

and HRv are identical.

VII. RESULTS

A. Codes CP

In Tables I–III, we present codes CP which improve theminimum distance by one compared to the codes with thesame length and dimension previously given in [6]. Together

4Notice however that CPv � CPu since C⊥Pu � C⊥

Pv when CPu and CPv aredefined by Gu(x) and Gv (x) respectively.

TABLE VII



WHERE α IS A ROOT OF x4 − x3 − 1

with the codes we list the corresponding Goppa polynomials asproduct of irreducible polynomials. The short optimal codesof the form [nμ = mdμ + dμ − 1, mdμ, dμ]q whose paritycheck matrices HKμ are used in the construction of codes CP

can be chosen to be MDS codes. The dimensions of the codes


TABLE VIII



WHERE α IS A ROOT OF x4 − x3 − 1 (CONTINUED)

in Tables I–III, are obtained by expressing their respectiveparity check matrices in reduced echelon form while theirminimum distances are obtained from the lower bound inTheorem 4.

B. Codes From CR

Using Construction X on the codes CRu and CRv as definedin Section VI and short optimal auxiliary codes, we are ableto obtain further improvements to the tables in [6] for F4,F7, F8, and F9. The minimum distance is again improved byone compared to the previous entries in [6]. The data of thecodes are shown in Table IV–VIII. The first column lists the

parameters of the resulting code, the two codes CRu and CRv,and the short auxillary codes. For the codes CRu and CRv, theGoppa polynomials are given in the second column.

APPENDIX

PARITY CHECK MATRIX OF A SUBFIELD SUBCODE

A subfield subcode C|Fq of a code C defined in Fqm consistsof all those codewords in C that have all their elements in thesubfield Fq . It is possible to construct the parity check matrixof a subfield subcode from the parity check matrix of thecode C defined in Fqm . The map πm is defined as,

πm : Fqm → Fmq

πm(β) = (a0, a1, . . . , am−1)T, β ∈ Fqm , ai ∈ Fq ,

(where T is the transpose operator) which maps elements ofFqm to the vector space F

mq using a suitable basis. Suppose H

is the parity check matrix of the code C in Fqm ,

H =

⎡

⎢⎢⎢⎣

h0,0 h0,1 · · · h0,n−1h1,0 h1,1 · · · h1,n−1

......

. . ....

hr−1,0 hr−1,1 · · · hr−1,n−1

⎤

⎥⎥⎥⎦

with redundancy r = n − k, then the parity check matrix ofthe subfield subcode C|Fq is given [9], by,

H̃ =

⎡

⎢⎢⎢⎣

πm(h0,0

) · · · πm(h0,n−1

)

πm(h1,0

) · · · πm(h1,n−1

)

.... . .

...πm

(hr−1,0

) · · · πm(hr−1,n−1

)

⎤

⎥⎥⎥⎦.

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous reviewersfor their helpful and constructive contributions to the paper.

REFERENCES

[1] V. D. Goppa, “A new class of linear error correcting codes,” Probl.Peredachi Inform., vol. 6, pp. 24–30, Sep. 1970.

[2] V. D. Goppa, “Rational representation of codes and (L , g)-codes,” Probl.Peredachi Inform., vol. 7, pp. 41–49, Sep. 1971.

[3] V. D. Goppa, “Codes constructed on the base of (L , g)-codes,” Probl.Peredachi Inform., vol. 8, no. 2, pp. 107–109, 1972.

[4] Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa, “Furtherresults on Goppa codes and their applications to constructing efficientbinary codes,” IEEE Trans. Inf. Theory, vol. 22, no. 5, pp. 518–526,Sep. 1976.

[5] M. Tomlinson, M. Jibril, C. Tjhai, S. Bezzateev, M. Grassl, andM. Z. Ahmed, “A generalized construction and improvements on nonbi-nary codes from Goppa codes,” IEEE Trans. Inf. Theory, vol. 59, no. 11,pp. 7299–7304, Nov. 2013.

[6] M. Grassl. (2007). Bounds on the Minimum Distance of Linear Codesand Quantum Codes [Online]. Available: http://www.codetables.de

[7] N. J. A. Sloane, S. M. Reddy, and C.-L. Chen, “New binary codes,”IEEE Trans. Inf. Theory, vol. 18, no. 4, pp. 503–510, Jul. 1972.

[8] K. K. Tzeng and K. Zimmermann, “On extending Goppa codes to cycliccodes (Corresp.),” IEEE Trans. Inf. Theory, vol. 21, no. 6, pp. 712–716,Nov. 1975.

[9] F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North Holland, 1983.

[10] W. Bosma, J. Cannon, and C. Playoust, “The MAGMA algebra system I:The user language,” J. Symbol. Comput., vol. 24, nos. 3–4, pp. 235–265,1997.


Mubarak Jibril received a Bachelor of Engineering in Electrical andElectronic Engineering from Bayero University Kano in 2005. He completeda Master of Science in Digital Communications and Signal Processing atthe University of Plymouth in 2008. He then completed a PhD in DigitalCommunications in 2011 from the University of Plymouth. He is currentlyemployed by the Nigerian Communications Satellite Company Limited,Abuja, Nigeria. His research interests include algebraic error correcting codes.

Sergey V. Bezzateev received the Diploma in Computer Science fromAerospace Instrumentation Institute of Leningrad, Soviet Union in 1980.In1987 he received Ph.D. degree in Information Theory from the AerospaceInstrumentation Institute of Leningrad. From 1980 till 1993 he was employedby Aerospace Instrumentation Institute. He spent 1993-1995 as researcher atthe Nagoya University, Japan, where he worked on Prof. Yoshihiro IwadareLaboratory. From 1995 he was Associate Professor at Department of Infor-mation technologies and Information Security, State University of AerospaceInstrumentation (SUAI), Saint Petersburg, Russia. From 2004 till 2007 he wasProject Leader in Joint Laboratory Samsung-SUAI on Information Security inWireless Networks. In 2010, he became Professor and the head of Departmentof Technologies of Information Security in SUAI. His main research interestsinclude coding theory and cryptography.

Martin Tomlinson received the BSc degree from the University of Birming-ham, UK in 1967 and the PhD in adaptive equalisation for data transmissionfrom the University of Loughborough in 1970. He is best known for the inven-tion of the Tomlinson-Harashima precoding technique. He worked at PlesseyTelecommunications Research Ltd, UK until 1975 in digital communicationsand satellite transmission. He then spent seven years with the UK Ministry ofDefence in the Satellite Communications Division of RSRE where he workedon the Skynet satellite, space and ground segments. He was project managerfor the communications payload of the NATO IV satellite before joiningthe University of Plymouth as the Head of the Communication EngineeringDepartment in 1982 where he became a Professor in 1987. In 2005 he wasa Visiting Erskine Fellow at the University of Canterbury, NZ. Dr Tomlinsonis currently Head of Fixed and Mobile Communications Research at theUniversity of Plymouth, leading research projects in satellite communications,coding, signal processing, wireless, encryption and information theoreticsecrecy which are his current research areas. He is also CEO of SRD WirelessLtd, a start up company specialising in applications of public key encryption.He has published over 200 papers in the fields of Digital Modulation andCoding, Signal Processing, Video Coding and Satellite Communications, aswell as contributing towards various satellite and wireless standards. ProfessorTomlinson has filed over 70 patents and is a member of the IET, a memberof the IEEE Signal Processing Society, a member of the IEEE InformationTheory Society, a member of the IEEE Communications Society and amember of the IEEE Satellite and Space Communications Society.

Markus Grassl (S’94–M’00) received his diploma degree in ComputerScience in 1994 and his doctoral degree in 2001, both from the Fakultätfür Informatik, Universität Karlsruhe (TH), Germany. His dissertation was onconstructive and algorithmic aspects of quantum error-correcting codes.

From 1994 to 2007 he was a member of the Institut für Algorithmenund Kognitive Systeme, Fakultät für Informatik, Universität Karlsruhe (TH),Germany. From 2007 to 2008 he was with the Institute for Quantum Opticsand Quantum Information of the Austrian Academy of Sciences in Innsbruck.From 2009 to 2014, he was a Senior Research Fellow at the Centre forQuantum Technologies at the National University of Singapore. In 2014,Markus Grassl joined the Institute of Optics, Information and Photonics at theUniversität Erlangen-Nürnberg and the Max Planck Institute for the Scienceof Light.

His research interests include quantum computation, focusing on quantumerror-correcting codes, and methods of computer algebra in algebraic codingtheory. He maintains tables of good block quantum error-correcting codes aswell as good linear block codes.

Mohammed Zaki Ahmed received his BEng (Hons) in CommunicationEngineering in 1999 and his doctoral degree in 2003, both from PlymouthUniversity. From 2001 to 2008 he was a lecturer at Plymouth University,from 2008 to 2010 he was a Senior Lecturer, and from 2010 an AssociateProfessor in Information Technology. In 2005 he was a research fellow of theJapanese Society for the Promotion of Science (JSPS). He has filed 9 patentsand has published more than 50 papers. His research interests include errorcorrection code design, and communication engineering and is currently thelead academic in algebraic coding at Plymouth University.

A Generalized Construction of Extended Goppa Codes

Documents

Transcript of A Generalized Construction of Extended Goppa Codes