On cyclic codes over finite chain rings

Abstract

Recent studies involve various approaches to establish a generating set for cyclic codes of arbitrary length over the class of Galois rings. One such approach involves the use of polynomials with minimal degree corresponding to specific subsets of the code, defined progressively. In this paper, we extend this approach to obtain a set of generators of cyclic codes over finite chain rings. Further, we observe that this set acts as a minimal strong Gröbner basis(MSGB) for the code.

MDS and MHDR cyclic codes over finite chain rings

“…We make some advancements to this study by establishing a unique set of generators for a cyclic code over a finite chain ring with arbitrary parameters. It is noted that this unique set of generators retains all the properties of generators obtained in [17].…”

“…The following results are straight forward generalisations of [16] for cyclic codes over the class of Galois rings to finite chain rings and have been communicated in [18]. These results are required to proceed further.…”

“…In this section, a unique set of generators for a cyclic code C of arbitrary length n over R has been established. For this, let us first recall the construction given by Monika et al to obtain a generating set for a cyclic code C over a finite chain ring R [17]. Let f 0 (z), f 1 (z), • • • , f m (z) be minimal degree polynomials of certain subsets of C such that deg f j (z) = t j < n and leading coefficient of f j (z) equal to γ i j u j , where u j is some unit in R, t j < t j+1 , i j > i j+1 and i j is the smallest such power.…”

“…Cyclic codes over rings have gained a lot of importance after the remarkable breakthrough given by Calderbank et al in [1]. A vast literature is available on the structure of cyclic codes over fields, integer residue rings, Galois rings and finite chain rings [2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,10,19,20,21,22,23,24] . Cyclic codes over finite chain rings with length coprime to the characteristic of residue field have been inestigated in [2,10,13].…”

“…They have also extended this approach over the finite chain ring F q + uF q + • • • + u k−1 F q−1 , u k = 0 [20]. Monika et al have given a constructive approach to establish a generating set for a cyclic code over a finite chain ring by making use of minimal degree polynomials of certain subsets of the code [17]. We make some advancements to this study by establishing a unique set of generators for a cyclic code over a finite chain ring with arbitrary parameters.…”