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].

For a given binary BCH code Cn of length n = 2s - 1 generated by a polynomial of degree r there is no binary BCH code of length (n + 1)n generated by a generalized polynomial of degree 2r. However, it does exist a binary cyclic code C(n+1)n of length (n + 1)n such that the binary BCH code Cn is embedded in C(n+1)n. Accordingly a high code rate is attained through a binary cyclic code C(n+1)n for a binary BCH code Cn. Furthermore, an algorithm proposed facilitates in a decoding of a binary BCH code Cn through the decoding of a binary cyclic code C(n+1)n, while the codes Cn and C(n+1)n have the same minimum hamming distance.

The BCH code C (respectively, C) of length n over a local ring Z pk (respectively, ℤp) is an ideal in the ring (Equation Presented) (respectively, (Equation Presented) which is generated by a monic polynomial that divides Xn - 1. Shankar [12] has shown that the roots of Xn - 1 are the unit elements of a suitable Galois ring extension GR(pk,s) (respectively, Galois field extension GF(p, s)) of the ring ℤpk (respectively, ℤp), where s is the degree of basic irreducible polynomial f(X) ∈ ℤpk [X]. In this study we assume that for st = bi, where 6 is prime and t is a non negative integer such that 0 ≤ i ≤ t, there exist corresponding chain of Galois ring extensions GR(pk, s,) (respectively, a chain of Galois field extensions GF(p, s,)) of ℤpk (respectively, ℤp), there are two situations; st = bi for i = 2 or st = bi for i ≥ 2. Consequently, the case is alike [12] and we obtain a sequence of BCH codes C0,C1, ···, Ct-1, C over ℤpk and C′0,C′1,···, C′t-1,C′ over ℤp with lengths n 0,n1,···, nt-1,n t. In second phase we extend the Modified Berlekamp-Massey Algorithm for the chain of Galois rings in such a way that the error will be corrected of the sequence of codewords from the sequence of BCH codes C0,C 1, ···, Ct-1,C. © Global Publishing Company.

In this paper we present matrices over unitary finite commutative local rings connected through an ascending chain of containments, whose elements are units of the corresponding rings in the chain such that the McCoy ranks are the largest ones.

This study establishes that for a given binary BCH code C0 n of length n generated by a polynomial g(x) ∈ F2[x] of degree r there exists a family of binary cyclic codes {Cm 2m−1(n+1)n}m≥1 such that for each m ≥ 1, the binary cyclic code Cm 2m−1(n+1)n has length 2m−1(n + 1)n and is generated by a generalized polynomial g(x 1 2m ) ∈ F2[x, 1 2m Z≥0] of degree 2mr. Furthermore, C0 n is embedded in Cm 2m−1(n+1)n and Cm 2m−1(n+1)n is embedded in Cm+1 2m(n+1)n for each m ≥ 1. By a newly proposed algorithm, codewords of the binary BCH code C0 n can be transmitted with high code rate and decoded by the decoder of any member of the family {Cm 2m−1(n+1)n}m≥1 of binary cyclic codes, having the same code rate.

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A0 ⊂ A1 ⊂···⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0 ⊂ K1 ⊂···⊂ Kt−1 ⊂ Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A∗ i and K∗ i are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A∗ i and K∗ i for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.