A Note on Linear Codes over Semigroup Rings

Computational & Applied Mathematics, 2005

Goppa and Srivastava codes over arbitrary local finite commutative rings with identity are constructed in terms of parity-cleck matrices. An efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, is proposed for Goppa codes.

Computers & Mathematics with Applications, 2011

For any finite commutative ring B with an identity there is a strict inclusion B[X ; Z 0 ] ⊂ B[X ; 1 2 ; Z 0 ] ⊂ B[X ; 1 2 2 Z 0 ] of commutative semigroup rings. This work is a continuation of Shah et al. (2011) [8], in which we extend the study of Andrade and Palazzo (2005) [7] for cyclic codes through the semigroup ring B[X ; 1 2 ; Z 0 ]. In this study we developed a construction technique of cyclic codes through a semigroup ring B[X ; 1 2 2 Z 0 ] instead of a polynomial ring. However in the second phase we independently considered BCH, alternant, Goppa, Srivastava codes through a semigroup ring B[X ; 1 2 2 Z 0 ]. Hence we improved several results of Shah et al. (2011) [8] and Andrade and Palazzo (2005) [7] in a broader sense.

Mathematical Sciences, 2012

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A 0 ⊂ A 1 ⊂ ⋯ ⊂ A t − 1 ⊂ A t be a chain of unitary commutative rings, where each A i is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K 0 ⊂ K 1 ⊂ ⋯ ⊂ K t − 1 ⊂ K t (another chain of unitary commutative rings), where each K i 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 A i and K i , 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 erro...

Computational & Applied Mathematics, 2011

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].

IEEE Transactions on Information Theory, 1997

In this correspondence we present a decoding procedure for Reed-Solomon (RS) and BCH codes defined over an integer residue ring pg q , where q is a power of a prime p: The proposed decoding procedure, as for RS and BCH codes over fields, consists of four major steps: 1) calculation of the syndromes; 2) calculation of the "elementary symmetric functions," by a modified Berlekamp-Massey algorithm for commutative rings; 3) calculation of the error location numbers; and 4) calculation of the error magnitudes. The proposed decoding procedure also applies to the synthesis of a shortest linear-feedback shift register (LFSR), capable of generating a prescribed finite sequence of elements lying in a commutative ring with identity.

Anais de XXX Simpósio Brasileiro de Telecomunicações, 2012

For a non negative 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 suitable Galois rings with multiplicative group A * i of units, and K0 ⊂ K1 ⊂ • • • ⊂ Kt−1 ⊂ Kt be the corresponding chain of unitary commutative rings, where each Ki is constructed by the direct product of corresponding residue fields of given Galois rings, with multiplicative groups K * i of units. This correspondence presents four different type of construction techniques of generator polynomials of sequences of BCH codes having entries from A * i and K * i for each i, where 0 ≤ i ≤ t. The BCH codes constructed in [1] are limited to given code rate and error correction capability, however, proposed work offers a choice for picking a suitable BCH code concerning code rate and error correction capability.

International Journal of Apllied Mathematics, 2014

A Goppa code is described in terms of a polynomial, known as Goppa polynomial, and in contrast to cyclic codes, where it is difficult to estimate the minimum Hamming distance d from the generator polynomial. Furthermore, a Goppa code has the property that d ≥ deg(h(X)) + 1, where h(X) is a Goppa polynomial. In this paper, we present a decoding principle for Goppa codes constructed by generalized polynomials, which is based on modified Berlekamp-Massey algorithm.

International Journal of Advanced Computer Science and Applications, 2015

In this paper, we study the structure of cyclic, quasi cyclic, constacyclic codes and their skew codes over the finite ring R. The Gray images of cyclic, quasi cyclic, skew cyclic, skew quasi cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error correcting codes are obtained from both cyclic and negacyclic codes over R. Some examples are given. Firstly, quasi constacyclic and skew quasi constacyclic codes are introduced. By giving two inner product, it is investigated their duality. A sufficient condition for 1 generator skew quasi constacyclic codes to be free is determined.

Information and Control, 1972

Given an integer m which is a product of distinct primes Pi, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF(pi). Specifically, if we are given a cyclic (n, ki) code over GF(pt) with minimum Hamming distance di, for each i, then we construct a code of block length n over the integers modulo m with 1-[~ p~i codewords, which is both linear and cyclic and has minimum Hamming distance mini di. i j k

2015

Let B be any finite commutative ring with identity. In this case, • • • ⊂ B[X; 1 p k Z 0 ] • • • ⊂ B[X; 1 p 2 Z 0 ] ⊂ B[X; 1 p Z 0 ], where p is a prime number and k ≥ 1, is the descending chain of commutative semigroup rings. All these semigroup rings are containing the polynomial ring B[X; Z 0 ]. In this paper, we introduced a construction technique of cyclic codes through the semigroup ring B[X; 1 p k Z 0 ] instead of a polynomial ring.

In this work we present a decoding procedure of BCH codes over local finite commutative rings with identity based on the Peterson-Gorenstein-Zierler algorithm.

Discrete Mathematics, 1997

It is well known that cyclic linear codes of length n over a (finite) field F can be characterized in terms of the factors of the polynomial x"-1 in F[x]. This paper investigates cyclic linear codes over arbitrary (not necessarily commutative) finite tings and proves the above characterization to be true for a large class of such codes over these rings. (~

IEEE Transactions on Information Theory, 2000

In this paper we study generalized Reed-Solomon codes (GRS codes) over commutative, noncommutative rings, show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context and we investigate their complexities. Under some hypothesis, the study of noncommutative generalized Reed-Solomon codes over finite rings leads to the fact that GRS code over commutative rings have better parameters than their noncommutative counterparts. Also GRS codes over finite fields have better parameters than their commutative rings counterparts. But we also show that given a unique decoding algorithm for a GRS code over a finite field, there exists a unique decoding algorithm for a GRS code over a truncated power series ring with a better asymptotic complexity. Moreover we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is for example a Galois ring or a truncated power series ring or the ring of square matrices over the latter ring.

Mathematical Sciences and Applications E-Notes, 2017

In this study, certain matrices are obtained using the elements of a finite chain ring. Then using these matrices as generator matrices; certain codes and their duals are obtained. Moreover relations between these codes, binary codes and Hadamard codes are explained.

Axioms

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.