Narrow Results

In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring ${\mathbb{Z}}_{2^m}$, with $m\ge 1$. We show that there exist two sequences ${\{C_{b^{j}n}\}}_{j\ge 1}$ and ${\{C_{b^{j}n}^{\prime }\}}_{j\ge 1}$ of non-primitive BCH codes (over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_{2^m}$, respectively) against primitive BCH codes $C_n$ of length $n$ and $C_n^{\prime }$ (over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_{2^m}$), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of $C_n$, $C_n^{\prime }$, ${\{C_{b^{j}n}\}}_{j\ge 1}$ and ${\{C_{b^{j}n}^{\prime }\}}_{j\ge 1}$. The selection of the schemes is subject to the choice of better code rate or better error-correction capability of the code. Finally, we present a decoding procedure for BCH codes over Galois rings, which is also used for the decoding of BCH codes over Galois fields, based on the modified Berlekamp–Massey algorithm.

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

Let R be an associative ring with identity and Z^*(R) be its set of non-zero zero-divisors. The undirected zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the non-zero zero-divisors of R, and where two distinct vertices r and s are adjacent if and only if rs = 0 or sr = 0. The distance between vertices a and b is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the superimum of these distances. In this paper, first we prove some results about Γ(R) of a semi-commutative ring R. Then, for a reversible ring R and a unique product monoid M, we prove $0\le \text{ diam}(\Gamma (R))\le \text{ diam}(\Gamma (R[M]))\le 3$. We describe all the possibilities for the pair diam(Γ(R)) and diam(Γ(R[M])), strictly in terms of the properties of a ring R, where R is a reversible ring and M is a unique product monoid. Moreover, an example showing the necessity of our assumptions is provided.

Let B[X; S] be a monoid ring with any fixed finite unitary commutative ring B and is the monoid S such that b = a + 1, where a is any positive integer. In this paper we constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes through monoid ring . For a = 1, almost all the results contained in [16] stands as a very particular case of this study.