Linear Codes over Finite Rings

Abstract

Abstract. In this paper we present a construction technique of cyclic, BCH, alternat, Goppa and Srivastava codes over a local finite commutative rings with identity.

A Note on Linear Codes over Semigroup Rings

“…In this section, we construct a subclass of alternant codes through a semigroup ring instead of a polynomial ring, which is similar to one initiated by Andrade and Palazzo [1] through polynomial rings. Goppa codes are described in terms of a Goppa polynomial h(X) and in contrast to cyclic codes, where it is difficult to estimate the minimum Hamming distance d from the generator polynomial, Goppa codes have the property that d ≥ deg(h(X)) + 1.…”

“…In [1] Andrade and Palazzo discussed the BCH, alternant, Goppa and Srivastava codes through the polynomial ring B[X; Z 0 ], where B is finite commutative ring with identity and Z 0 = Z + ∪ {0}. In [2] T. Shah et.…”

“…considered linear codes over the semigroup ring B[X; 1 2 Z 0 ]/(X n − 1). In this paper, we introduce construction techniques of these codes through the semigroup ring B[X; 1 3 Z 0 ] instead of the polynomial ring B[X; Z 0 ], where we improve the results of [1].…”

“…In this work we take B as a finite commutative ring with unity and in the same spirit of [1], we fix a cyclic subgroup of group of units of the ring B[X; This paper is organized as follows. In Section 2, we give some basic results on semigroups and semigroup rings necessary for the construction of the codes.…”

“…Z0] instead of the polynomial ring B[X; Z0], where B is a finite commutative ring with identity, and for these constructions we improve the several results of [1]. After this, we present a decoding principle for BCH, alternant and Goppa codes which is based on modified Berlekamp-Massey algorithm.…”