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, Mathematical subject classification: 18B35, 94A15, 20H10.

Abstract. In this paper, we introduced new construction techniques of BCH, alternant, Goppa, Srivastava codes through the semigroup ring B[X;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. This algorithm corrects all errors up to the Hamming weight t ≤ r/2, i.e., whose minimum Hamming distance is r + 1.

Let B be any finite commutative ring with identity.where a ∈ {2, 3, 5, 7, • • •}, k ≥ 1, is the descending chain of commutative semigroup rings. All these semigroup rings are containing the polynomial ring B[X; Z0]. In this paper initially we introduced the construction technique of cyclic codes through a semigroup ring B[X; 1 a k Z0] instead of a polynomial ring. After this we separately considered BCH, alternant, Goppa, Srivastava codes and by this new constructions we improve the several results of [1] by adopting the same lines as in [1].

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