Chain of finite rings and construction of BCH Codes

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...

Linear Algebra and its Applications, 1999

BCfi wdes over arbitrary finite ~~~nirnut:!tiv~ rings with identity arc drip& in LCF~S of their locator vector, The derivation is hased on the factorization of .I-' --I over the unit ring of an ~tppropr~lt~ extension of the finite rin g. We prcscnt an ~~~ci~nt,d~~iu~ procedure, based on the modified Berlekamp Massey ;li~~~rithrn. for that codes. The code construction and the decoding proccdurcs arc very similar to the BCH codes over finite integer rings. 43 1999 Ekxvier Scicncc Inc. Ail rights rwrwd. t f MS ~icl.Ev~~~~t~~it, 94BM: 94?35 Linear codes over rings have recently r&xd a great interest for their new role in algebraic coding theory and for their successful application in combined *Corr~s~ndin~ author. E-m& andr~detrt:mat.ithiI~~.unusy.hr, '

Computational & Applied Mathematics, 2003

Alternant codes over arbitrary finite commutative local rings with identity are constructed in terms of parity-check matrices. The derivation is based on the factorization of x s − 1 over the unit group of an appropriate extension of the finite ring. An efficient decoding procedure which makes use of the modified Berlekamp-Massey algorithm to correct errors and erasures is presented. Furthermore, we address the construction of BCH codes over Z m under Lee metric.

International Journal of Algebra, 2014

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

In this work, we introduce a method by which it is established that how a sequence of non-primitive BCH codes can be obtained by a given primitive BCH code. For this, we rush to the out of routine assembling technique of BCH codes and use the structure of monoid rings instead of polynomial rings. Accordingly, it is gotten that there is a sequence {C b j n } 1≤ j≤m , where b j n is the length of C b j n , of non-primitive binary BCH codes against a given binary BCH code C n of length n. Matlab based simulated algorithms for encoding and decoding for these type of codes are introduced. Matlab provides in routines for construction of a primitive BCH code, but impose several constraints, like degree s of primitive irreducible polynomial should be less than 16. This work focuses on non-primitive irreducible polynomials having degree bs, which go far more than 16.

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.

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.

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].

TEMA - Tendências em Matemática Aplicada e Computacional, 2005

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

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.