An Explicit Construction of -Generator Quasi-Twisted Codes

5770 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 12, DECEMBER 2008

[21] W. M. Kantor, “Commutative semifields and symplectic spreads,” J. ofAlgebra, vol. 270, pp. 96–114, 2003.

[22] W. M. Kantor, “Finite semifields,” in Finite Geometries, Groups, andComputation (Proc. Conf. Pingree Park, CO Sep. 2005), Berlin, Ger-many, Sept. 2006, pp. 103–114, de Gruyter.

[23] Y. Luo, F.-W. Fu, A. J. Han Vinck, and W. Chen, “On constant com-position codes over � ,” IEEE Trans. Inf. Theory, vol. 49, no. 11, pp.3010–3016, Nov. 2003.

[24] G. Myerson, “Period polynomials and Gauss sums for finite fields,”Acta. Arithmetica, vol. XXXIX, pp. 251–264, 1981.

[25] N. V. Semakov and V. A. Zinoviev, “Equidistant q-ary codes withmaximal distance and resolvable balanced incomplete block designs,”Probl. Inf. Transm., vol. 4, no. 2, pp. 1–7, 1968.

[26] N. V. Semakov, V. A. Zinoviev, and G. V. Zaitsev, “A class of maximalequidistant codes,” Probl. Inf. Transm., vol. 5, no. 2, pp. 65–68, 1969.

[27] D. Slepian, “Permutation modulation,” Proc. IEEE, vol. 53, pp.228–236, 1965.

[28] T. Storer, Cyclotomy and Difference Sets. Chicago, IL: Markham,1967.

[29] M. Svanström, “Construction of ternary constant-composition codeswith weight three,” IEEE Trans. Inf. Theory, vol. 46, no. 7, pp.2644–2647, 2000.

[30] M. Svanström, P. R. J. Östergard, and G. T. Bogdanova, “Bounds andconstructions for ternary constant-composition codes,” IEEE Trans.Inf. Theory, vol. 48, no. 1, pp. 101–111, Jan. 2002.

[31] V. A. Zinoviev, “Cascade equal-weight codes and maximal packings,”Probl. Contr. Inf. Theory, vol. 12, no. 1, pp. 3–10, 1983.

An Explicit Construction of -GeneratorQuasi-Twisted Codes

Eric Z. Chen

Abstract—Quasi-twisted (QT) codes are a generalization of quasi-cyclic(QC) codes. Based on consta-cyclic simplex codes, a new explicit construc-tion of a family of �-generator quasi-twisted (QT) two-weight codes is pre-sented. It is also shown that many codes in the family meet the Griesmerbound and therefore are length-optimal. New distance-optimal binary QC�� , �� , and �� codes, and good ternary QC�� and �� codes are also obtained by the construction.

Index Terms—Linear codes, optimal codes, quasi-cyclic codes,quasi-twisted codes, simplex codes.

I. INTRODUCTION

As a generalization to cyclic codes, quasi-cyclic (QC) codes andquasi-twisted (QT) codes have been shown to contain many good linearcodes. Many researchers have been using modern computers to searchfor good QC or QT codes, and many record-breaking codes are found[1]–[12]. The problem with this method is that it becomes intractablewhen the dimension and the length of the code become large. Unfortu-nately, very little is known on explicit constructions of good QC or QTcodes. For �-generator QC or QT codes, even fewer results are known[13], [14].

Manuscript received June 12, 2008; revised August 28, 2008. Current ver-sion published November 21, 2008. The material in this correspondence waspresented in part at the IEEE ISIT2007, Nice, France, June 2007.

The author is with the Department of Computer Science, Kristianstad Uni-versity College, 291 88 Kristianstad, Sweden (e-mail: [email protected]).

Communicated by I. Dumer, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2008.2006420

A linear code is called projective if any two of its coordinates arelinearly independent, or in other words, if the minimum distance of itsdual code is at least three. A code is said to be two-weight if it has onlytwo nonzero weights. Projective two-weight codes are closely relatedto strongly regular graphs [15].

In this paper, a new explicit construction of a family of �-generatorQT two-weight codes is presented. It is the first time that a family of�-generator QT codes is constructed systematically. It is also shownthat many codes of this family are good and optimal. Examples aregiven to show the construction and the modular structure of the codes.

II. QUASI-TWISTED CODES AND TWO-WEIGHT CODES

A. Consta-Cyclic Codes

The codes discussed in the following sections are linear. A �-arylinear code is a �-dimensional subspace of an �-dimensional vectorspace over the finite field �� , with minimum distance d between anytwo codewords. We denote a �-ary code as an �� code, or a bi-nary �� code if � � �. A linear �� code is said to be�-consta-cyclic if there is a nonzero element � of �� such that forany codeword �� , a consta-cyclic shift by one posi-tion or �� is also a codeword [16]. Therefore, theconsta-cyclic code is a generalization of the cyclic code, and a cycliccode is a �-consta-cyclic code with � � �. A consta-cyclic code canbe defined by a generator polynomial.

B. Hamming Codes and Simplex Codes

Hamming codes are a family of linear single error correcting codes.For any positive integer � and prime power �, we have a Hamming�� code, where � � ��. Further, if and ��are relatively prime, then the Hamming code is equivalent to a cycliccode.

The dual code of a Hamming code is called the simplex code. Sofor any integer � and prime power �, there is a simplex �� code. It should be noted that a simplex code isan equidistance code, where �� nonzero codewords have a weightof ��. Let �� be a primitive polynomial of degree over �� . A�-consta-cyclic simplex �� code can be definedby the generator polynomial �� , where � � �� and � has order of � � � [16]. Further, a simplex code isequivalent to a cyclic code if and� � � are relatively prime.

C. Quasi-Twisted Codes

A code is said to be quasi-twisted (QT) if a consta-cyclic shift of anycodeword by p positions is still a codeword. Thus a consta-cyclic codeis a QT code with � � �, and a quasi-cyclic (QC) code is a QT codewith � � �. The length n of a QT code is a multiple of �, i.e., � � ��.

The consta-cyclic matrices are also called twistulant matrices. Theyare basic components in the generator matrix for a QT code. An��consta-cyclic matrix is defined as

� �

��

......

......

...��

(1)

and the algebra of � � � consta-cyclic matrices over �� is isomor-phic to the algebra in the ring � � �� if � is mapped onto thepolynomial formed by the elements of its first row, ��

��, with the least significant coefficient on the left. The

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polynomial c(x) is also called the defining polynomial of the matrix � .A twistulant matrix is called a circulant matrix if � � �.

The generator matrix of a QT code can be transformed into rows oftwistulant matrices by suitable permutation of columns. So a �-gener-ator QT code has the following form of the generator matrix [17]

� � �� (2)

where �� , are twistulant matrices of order �. A�-generator QT code has generator matrix of the following form

� ��

��

(3)

where�� are twistulant matrices, for � � �� , and � � �� .Very little on �-generator QT codes is known in the literature. Two

�-generator QC codes were given in [13], while a construction methodfor �-generator QC codes was presented in [14]. In both cases, thecodes were constructed by the help of modern computers. In the fol-lowing section, we will present an explicit construction of a familyof �-generator QT codes, that are also two-weight codes, and we willprove that for many parameters they are good or optimal.

D. Two-Weight Codes

Let � and �, be the two non-zero weights of a two-weight code,where � �� . We denote a projective -ary linear two weight codeas an �� code. There is an online database of two-weightcodes [20]. In the survey paper [15], Calderbank and Kantor listedmany known families of projective two-weight codes. Among thosefamilies, there is a family of two-weight �� codes, denotedby SU2. For any prime power q, positive integers � � and i, the codeshave the following parameters:

Block length � � ��

Dimension � � �

Weights � � ��

where � � � � �. No explicit construction of the code is known andstudied in the literature, and very little is known about its structure andperformance.

III. �-GENERATOR QUASI-TWISTED TWO-WEIGHT CODES

A. Explicit Construction

For any integer � � and prime power , a �-consta-cyclicsimplex �� code can be constructed. Let�� be the generator polynomial for the �-consta-cyclic simplexcode. Let �� be � � nonzero elements of �� and let� � � � �� . Any codeword of the �-consta-cyclic simplex�� code can be expressed by a polynomial��

��, with the computation modulo �� , where� � �� , and � � �� . Let �� be the twistulantmatrix defined by the generator polynomial ��, and �� be thetwistulant matrix defined by ��. So totally, we obtain � � �twistulant matrices. We construct a generator matrix of a �-generatorQT code as follows:

� ��

� ��

��

��

(4)

where �� are � � � different twistulant matrices fromthe � � � twistulant matrices �� and� � �� defined above, �� and �� are the first and

second row of the twistulant matrices of �. Then we have followingresult:

Theorem 1: For any positive integer � �, and prime power ,the generator matrix given in (4) defines a �-generator QT two-weight�� code, where � � �� and� � �� .

Proof: A �-consta-cyclic simplex �� code is an equidistance code. Its � � � nonzero codewords can bepartitioned into q–1 groups of codewords, such that each group has� � � � �� codewords, and these codewords are �-consta-cyclic each other. Let m(x) be the message polynomial to be encodedby a defining polynomial ��

��. Let �� and ��be two polynomials corresponding to the distinct twistulant matrices�� and ��. Then the codewords obtained by encoding m(x) by ��and ��, i.e., �� and �� , will be different, wherethe multiplication is computed with modulo �� . Let �� be thesub-code defined by ��, and �� be the sub-code defined by ��. So�� consists of codewords that just repeat the codewords of the consta-cyclic simplex �� code by p times. Therefore, �� is also anequidistance code with a distance of ��. Similarly, �� is also anequidistance code with a distance of � � ��, since its codewordconsists of first m zeros, followed by � � �� different codewords ofthe consta-cyclic simplex �� code. Based on the equidistanceproperty of the simplex code and the generator matrix structure of (4),the sum of nonzero codewords from �� and �� has a weight of � ��, or ��, depending on the codewords in �� and �� havethe same codeword from the consta-cyclic simplex �� codeor not. Therefore, any nonzero codeword of the �-generator QT codedefined by (4) has a weight � � � � �� or � � ��. Thisproves Theorem 1. Q.E.D.

Theorem 1 provides the complete solution to this family oftwo-weight codes. The results presented in [21] are two specialcases. We state them as corollaries. When � �, the binary simplex�� code is cyclic. So the constructed two-weight code isquasi-cyclic.

Corollary 2: Let � �. For any positive integer � �, the generatormatrix (4) defines a �-generator QC two-weight �� code, where � � �� .

If and � � are relatively prime, the simplex �� code is also a cyclic code, and thus we have the followingresult.

Corollary 3: Let � � be any positive integer, and q be a primepower. If and � � are relatively prime, the generator matrix (4)defines a �-generator QC two-weight �� code, where � � � � �� and � � �� .

B. Examples

Example 1: Let � � and � �. Then we have �� . So a binary cyclic simplex �� code canbe defined by �� . Let �� be the circulant matrixdefined by the generator polynomial ��, and �� be the circulantmatrices defined by �� , where � � �� . Thenthe following generator matrix defines a binary �-generator QC two-weight �� code:

� ��

� ��

�

If we take � columns of the matrices in the above generator matrix,we obtain other binary �-generator QC two-weight �� ,

5772 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 12, DECEMBER 2008

�� , �� , �� , �� , and�� codes in the series, for � � � � �.

Example 2: Let � � and � �. Then � �� and � � � �.Since � and � are relatively prime, we have a cyclic simplex �� code. It can be shown that �� defines a cyclic simplex �� code. So a series of�-generator QC two-weight �� can be constructed,with � �� .

Example 3: Let � � and � �. Then � �� , and �. Since � and � � � are not relatively prime, we have a �-consta-cyclic �� code. One primitive polynomial of degree � over � is�� . So the corresponding generator polynomial forthis �-consta-cyclic code is �� .Let � �� . Let � be the twistulant matrix defined by g(x),and �� be the twistulant matrices defined by ��, for � �� ,and � � �� and 3. With Theorem 1, a series of �-generator QTtwo-weight �� codes can be constructed for � �� .

IV. GOOD AND OPTIMAL CODES

A. Distance-Optimal Codes

A linear �� code is distance-optimal or �-optimal if itsminimum distance cannot be improved, i.e., if there does not exist an�� code. For given �, �, and �, there is an online table ofbest-known linear codes [18]. It provides the bound on the minimumdistance of a code. For some parameters, the bounds are exact, whilefor others, the lower and upper bounds are presented. A code that meetsthe bound is �-optimal. A code is said to be good if it reaches the lowerbound on the minimum distance, since no codes with larger distanceare known. For binary quasi-cyclic codes, there is an online databaseon best-known binary QC codes [19]. Among the �-generator QCand QT two-weight codes constructed above, many codes are good,and �-optimal. For example, the binary �-generator QC two-weight�� codes with � � � � �, and �� codes with � � � � �� are �-optimal. The �-generator QT two-weight�� code with � � � � �, the �-generator QC�� code with � � � � ��, and the �-generatorQT �� code with �� are d-optimal too.Among these codes, �� , �� , and �� codesare previously unknown to be quasi-cyclic [19]. By computer searchfor good codes, Gulliver and Bhargava constructed �-generator QC�� code [22]. But with � �� , and � �, a �-optimal�-generator QT �� code can be constructed by our method.With � �� , and � �� and � ��, �-generator QCtwo-weight �� and �� codes can beobtained and they reach the lower bound on the distance [18].

B. Length-Optimal Codes

A linear �� code is length-optimal or n-optimal if its lengthcannot be improved, i.e., if there does not exist an �� code.For an �� code, the block length n is ruled by Griesmer bound[16]

� �

��

��

�

��(5)

where �� denotes the smallest integer greater than or equal to �.Let � ��. Then a simplex code with dimension �� can be con-

structed in a �-generator QT form.

Theorem 4: For any prime power q and integer � � �, let � ��

and � �� . The following generator matrix defines a�-generator QT simplex �� code:

� ��

��

� (6)

This theorem can be proved similar to Theorem 1. So the proof isomitted.

The given examples show that many SU2 codes are good or distance-optimal. The following theorem tells how good the codes are in general.For integers � � �� , and � �� , we definea function called gap as follows:

� ��

�

��

�

�� (7)

Theorem 5: For prime power q, integer � � �, let i, r, and p beintegers such that � �� , and � �� . The �-generator QT �� code generated by the generator matrix given in (4) and (6) meets theGriesmer bound with a gap given by gap�� .

Proof: By the Griesmer bound, the length n of a code of dimen-sion � ��, and distance � �� satisfies

� �

��

��

�

��

��

��

��

��

� �

��

��

��

��

��

��

��

��

� � ��

�

��

��

��

� � ��

� � ��

�

��

��

��

Since

�

��

��

��

�

��

��

�

��

�

��

so we have

� � ��

� ��

or

� � ��

So the code meets the Griesmer bound with a gap defined by (7).Q.E.D.

If � �� . So we have the following result.

Corollary 6: For prime power q and integer � � �, let � and � beintegers such that � �� , and � ��. The �� code constructed meets the Griesmerbound with equality, and thus is length-optimal.

As an example, let us consider � � �. Table I lists the calculatedresults on the codes obtained for � �� . It shows thatthe code becomes better when p increases. In the table, gb denotes theGriesmer bound on the length. But maximum p for a �-generator QT

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TABLE IEXAMPLE OF CODES

two-weight code in the family is ��, and when � � ��, a �-generatorQT simplex code is obtained.

ACKNOWLEDGMENT

The author would like to thank the anonymous referees and the As-sociate Editor for their helpful comments and suggestions that signifi-cantly improved the presentation of the results.

REFERENCES

[1] C. L. Chen and W. W. Peterson, “Some results on quasi-cyclic codes,”Inf. Contr., vol. 15, pp. 407–423, 1969.

[2] E. J. Weldon Jr., “Long quasi-cyclic codes are good,” IEEE Trans. Inf.Theory, vol. 13, pp. 130–, Jan. 1970.

[3] T. Kasami, “A gilbert-varshamov bound for quasi-cyclic codes of rate1/2,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 679–, 1974.

[4] S. Ling and P. Solé, “Good self-dual quasi-cyclic codes exist,” IEEETrans. Inf. Theory, vol. 39, pp. 1052–1053, 2003.

[5] H. C. A. van Tilborg, “On quasi-cyclic codes with rate 1/m,” IEEETrans. Inf. Theory, vol. IT-24, pp. 628–629, Sep. 1978.

[6] T. A. Gulliver and V. K. Bhargava, “Some best rate 1/p and rate�� systematic quasi-cyclic codes,” IEEE Trans. Inf. Theory,vol. IT-37, pp. 552–555, May 1991.

[7] E. Z. Chen, “Six new binary quasi-cyclic codes,” IEEE Trans. Inf.Theory, vol. IT-40, pp. 1666–1667, Sep. 1994.

[8] R. N. Daskalov, T. A. Gulliver, and E. Metodieva, “New good quasi-cyclic ternary and quaternary linear codes,” IEEE Trans. Inf. Theory,vol. 43, pp. 1647–1650, 1997.

[9] P. Heijnen, H. C. A. van Tilborg, T. Verhoeff, and S. Weijs, “Somenew binary quasi-cyclic codes,” IEEE Trans. Inf. Theory, vol. 44, pp.1994–1996, Sep. 1998.

[10] N. Aydin, I. Siap, and D. Ray-Chaudhury, “The structure of 1-generatorquasi-twisted codes and new linear codes,” Design, Codes Cryptogr.,vol. 24, pp. 313–326, 2001.

[11] R. Daskalov and P. Hristov, “New quasi-twisted degenerate ternarylinear codes,” IEEE Trans. Inf. Theory, vol. 49, pp. 2259–2263, 2003.

[12] R. Daskalov, P. Hristov, and E. Metodieva, “New mimimum distancebounds for linear codes over GF(5),” Discr. Math., vol. 275, pp.97–110, 2004.

[13] T. A. Gulliver and V. K. Bhargava, “Two new rate 2/p binary quasi-cyclic codes,” IEEE Trans. Inf. Theory, vol. 40, pp. 1667–1668, Sep.1994.

[14] E. Z. Chen, “New quasi-cyclic codes from simplex codes,” IEEE Trans.Inf. Theory, vol. 53, pp. 1193–1196, Mar. 2007.

[15] R. Calderbank and W. M. Kantor, “The geometry of two-weightcodes,” Bull. London Math. Soc., vol. 18, pp. 97–122, 1986.

[16] E. R. Berlekamp, Algebraic Coding Theory, Revised 1984 ed. :Aegean Park, 1984.

[17] G. E. Séguin and G. Drolet, “The theory of 1-generator quasi-cycliccodes” Dept. of Electr. Comp. Eng., Royal Military College of Canada,Kingston, ON, Canada, Jun. 1990, Manuscript.

[18] M. Grassl, Bounds on the Minimum Distance of Linear Codes [Online].Available: http://www.codetables.de

[19] E. Z. Chen, Web Database of Binary QC Codes [Online]. Available:http://moodle.tec.hkr.se/~chen/research/codes/searchqc2.htm

[20] E. Z. Chen, Web Database of Two-Weight Codes [Online]. Available:http://moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php

[21] E. Z. Chen, “New constructions of a family of 2-generator quasi-cyclictwo-weight codes and related codes,” in Proc. 2007 IEEE Int. Symp. In-formation Theory (ISIT2007), Nice, France, Jun. 2007, pp. 2191–2195.

[22] T. A. Gulliver and V. K. Bhargava, “Some best rate 1/p and rate�� systematic quasi-cyclic codes over GF(3) and GF(4),” IEEETrans. Inf. Theory, vol. 38, no. 4, pp. 1369–1374, Jul. 1992.

Chain of Separable Binary Goppa Codes and TheirMinimal Distance

Sergey Bezzateev and Natalia Shekhunova

Abstract—It is shown that subclasses of separable binary Goppa codes,�� - codes, with � � �� GF�� and special Goppapolynomials �� can be presented as a chain of embedded and equivalentcodes. The true minimal distance has been obtained for all codes of thechain.

Index Terms—Goppa codes, quasi-cyclic Goppa codes, minimal distanceof separable binary codes.

I. INTRODUCTION

Any �-any �� Goppa code can be defined by two objects:Goppa polynomial��where�� is a polynomial of a degree � overGF�� and location set � where � � �� GF�� .

Definition 1: A �-ary vector � �� of a length where is a cardinality of the set � � �� GF�� is acodeword of the �� code if and only if the following equation issatisfied:

�

��

��

� � ��

The parity-check matrix of the �� code can be presented in thefollowing form:

� �

��

��

�

��

��

� � ��

��

�

��

�

It is known that the �� code has following parameters [1]:• the length of the code is equal to the cordinality of the location

set �� ;• the dimension is � ��; and• the true minimal distance is � � � �.

The �� code is a binary Goppa code if � � �. The �� codeis a separable Goppa code if all roots of its Goppa polynomial ��

Manuscript received June 12, 2008; revised August 28, 2008. Current versionpublished November 21, 2008.

The authors are with the Department of Information Systems and Security,Saint Petersburg State University of Airspace Instrumentation, Saint-Petersburg,Russia (e-mail: [email protected]; [email protected]).

Communicated by G. Seroussi, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2008.2006420

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An Explicit Construction of -Generator Quasi-Twisted Codes

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