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Encoding Through Generalized Polynomial Codes “main” — 2011/7/8 — 10:21 — page 349 — #1 Volume 30, N. 2, pp. 349–366, 2011 Copyright © 2011 SBMAC ISSN 0101-8205 www.scielo.br/cam Encoding through generalized polynomial codes T. SHAH1, A. KHAN1 and A.A. ANDRADE2∗ 1Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 2Department of Mathematics, Ibilce, Unesp, São José do Rio Preto, SP, Brazil E-mails: [email protected] / [email protected] / [email protected] Abstract. This paper introduces novel constructions of cyclic codes using semigroup rings instead of polynomial rings. These constructions are applied to define and investigate the BCH, alternant, Goppa, and Srivastava codes. This makes it possible to improve several recent results due to Andrade and Palazzo [1]. Mathematical subject classification: 18B35, 94A15, 20H10. Key words: semigroup ring, cyclic code, BCH code, alternant code, Goppa code, Srivastava code. 1 Introduction In ring theory, finite commutative rings are of interest due to many applica- tions. The role of ideals is very essential for these applications and it is often important to know when the ideals in a ring are principal ideals. The very fa- mous class of rings in this regard is the polynomial rings in one indeterminate coefficients from a finite field, in fact it is an Euclidean domain. The codingfor error control has a vital role in the design of modern communication systems and high speed digital computers. Most of the classical error-correcting codes are ideals in finite commutative rings, especially in quotient rings of Euclidean domains of polynomials and group rings, i.e., cyclic codes are principal ideals F /( n ) in the quotient ring q [X] X − 1 . #CAM-187/10. Received: 09/II/10. Accepted: 01/I/11. *Acknowledgment to FAPESP by financial support, 2007/56052-8. “main” — 2011/7/8 — 10:21 — page 350 — #2 350 ENCODING THROUGH GENERALIZED POLYNOMIAL CODES On the above ideas, Cazaran and Kelarev [2] established necessary and suffi- cient conditions for an ideal to have a single generator and described all finite Z , , / , quotient rings m[X1 ∙ ∙ ∙ Xn] I where I is an ideal generated by univariate polynomials which are commutative principal ideal rings. In another paper, Cazaran and Kelarev [3] obtained conditions for the certain rings to be finite commutative principal ideal rings. However, the extension of a BCH code C embedded in a semigroup ring F[S], where S is a finite semigroup, was con- sidered in 2006 by Cazaran et. all [4], where an algorithm was presented for computing the weights of extensions for these codes embedded in semigroup rings as ideals. A lot of information concerning various ring constructions and about polynomial codes is given by Kelarev [5]. In [5], the whole Sections 9.1 and 9.2 are reserved to error-correcting codes in ring constructions closely re- lated to semigroup rings. Especially, Section 9.1 deals error-correcting cyclic codes of length n which are ideals in group ring F[G], where F is a field and G is a finite torsion group of size n. Another work concerning extensions of BCH codes in various ring constructions has been given by Kelarev ([6, 7]), where the results can also be considered as the special cases of semigroup rings of specific nature. A.A. Andrade and R. Palazzo Jr. [1] discussed the cyclic, BCH, alternant, Goppa and Srivastava codes over finite rings, which are in fact constructed through a polynomial ring in one indeterminate with a finite coefficient ring. In this paper, we introduce the construction techniques of cyclic codes through a semigroup ring instead of a polynomial ring and then establish the construc- tions of BCH, alternant, Goppa, Srivastava codes. Here the results of [1] are improved in such a way that instead of cancellative torsion free additive monoid Z 1 Z 0, the cancellative torsion free additive monoid 2 0 is used which shifts whole construction of a finite quotient ring of a polynomial ring into afinite quotient ring of a semigroup ring of specific type. Furthermore, B is taken as a finite commutative ring with unity in the same spirit of [1]. A cyclicsub- 1 Z /( n ) group of group of units of the ring B[X; 2 0] X − 1 is fixed analogous 2s to [1]. In this set up the factorization of X − 1 over the group of units of 1 Z /( n ) B[X; 2 0] X − 1 is again a difficult task. The procedure adopted in this work for construction of linear codes through 1 Z the semigroup ring B[X; 2 0] is simple as polynomial’s set up and our ap- Comp. Appl. Math., Vol. 30, N. 2, 2011 “main” — 2011/7/8 — 10:21 — page 351 — #3 T. SHAH, A. KHAN and A.A. ANDRADE 351 proach is quite different to the embedding of linear polynomial codes in a semi- group ring or in a group algebra, which has been adopted by several authors. This paper is organized as follows. In Section 2, the basic results on semi- groups and semigroup rings necessary for the construction of the codes are given. In Section 3, the construction of cyclic codes through a semigroup ring is introduced. Section 4, addresses the constructions of BCH and alternant codes through the semigroup rings. In Section 5, a construction of Goppa and Srivas- tava codes through the semigroup rings is described. Finally, in Section 6, the concluding remarks are presented. 2 Preliminaries In this section, we review basic facts on commutative semigroup rings from ( , , ) ( , ) [8]. Assume that B + ∙ is an associative ring and S ∗ is a semigroup. Let J be the set of all finitely nonzero functions f from S into B. The set J is a ring with respect to binary operations addition and multiplication defined ( )( ) ( ) ( ) ( )( ) ( ) ( ), as f + g s = f s + g s and f g s = f t g u where the symbol t∗u=s indicates that the sum is taken over all pairsP (t, u) of elements of S such t∗u=s thatP t ∗ u = s and it is understood that in the situation where s is not express- , , ( )( ) ible in the form t ∗ u for any t u ∈ S then f g s = 0. The ring J is known as a semigroup ring of S over B. If S is a monoid, then J is called a monoid ring. This ring J is represented as B S whenever S is a multiplicative semi- [ ] n ( ) ( ) group and elements of J are written either as f s s or as f si si . s S i 1 P∈ P= The representation of J will be B[X; S] whenever S is an additive semigroup. As there is an isomorphism between additive semigroup S and multiplicative s , semigroup {X : s ∈ S} so a nonzero element f of B[X; S] is uniquely rep- resented in the canonical form n n ( ) si si , . f si X = fi X where fi 6= 0 and si 6= s j for i 6= j Xi=1 Xi=1 The concepts of degree and order are not generally defined in semigroup rings. If the semigroup S is a cancellative torsion free or totally ordered, we can define the degree and the order of an element of the semigroup ring B[X; S] Comp. Appl. Math., Vol. 30, N. 2, 2011 “main” — 2011/7/8 — 10:21 — page 352 — #4 352 ENCODING THROUGH GENERALIZED POLYNOMIAL CODES n si in the following manner; if fi X is the canonical form of a nonzero element i 1 , < P=< < f of R[X; S] where s1 s2 ∙ ∙ ∙ sn, then sn is called the degree of pseudo ( ) polynomial f and we write deg f = sn and similarly the order of f is written ( ) . , as ord f = s1 Now, if R is an integral domain, then for f g ∈ B[X; S], we have ( ) ( ) ( ) deg f g = deg f + deg g ( ) ( ) ( ). ord f g = ord f + ord g If the monoid S is Z0 and B is an associative ring, the semigroup ring J Z 1 Z is simply the polynomial ring, that is, B[X] = B[X; 0] ⊂ B[X; 2 0]. Further- 1 Z more in B[X; 2 0] one may define the degree of a pseudo polynomial because 1 Z 2 0 is totally ordered. In addition B[G] is known as group ring whenever G is a group. Particularly F[G] is group algebra, where F is a field. In [5] the Section 9.1 is dealing with error-correcting cyclic codes of length n which are ideals in group ring F[G], where G is taken to be a finite torsion group of size n. 3 Cyclic codes through a semigroup ring According to [9], if an ideal I of a commutative ring < with unity is generated , by an element a of < then in any quotient ring < of <, the corresponding ideal I is generated by the residue class a of a. Hence, every quotient ring of a principal ideal ring (PIR) is a PIR as well. It follows that the ring Zn is a PIR Fq [X;Z0] for any non prime positive integer n. Consequently the ring ( n ) , < = X −1 where q is a power of a prime p, is a PIR. Also, if q is a power of a prime p F 1 Z Zq [X;Z0] q [X; 2 0] then ( n ) is a PIR (see also [1]). By the same argument ( n ) < = X −1 < = X −1 Z 1 Z q [X; 2 0] and ( n ) are PIRs. Furthermore, the homomorphic image of a PIR < = X −1 is again a PIR [10, Proposition 38.4].
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