Sequences of primitive and non-primitive 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...

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 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.

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, '

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.

In this paper we propose a technique to blindly synthesize the generator polynomial of BCH codes. The proposed technique involves finding Greatest Common Divisor (GCD) among different codewords and block lengths. Based on this combinatorial GCD calculation, correlation values are found. For a valid block length, the iterative GCD calculation results either into generator polynomial or some of its higher order multiples. These higher order polynomials are factorized under modulo-2 operation, and one of the resulting factors is always the generator polynomial which further increases the correlation value. The resulting correlation plot for different polynomials shows very high values for correct block length and valid generator polynomial. Knowing the valid block length and generator polynomial, all other parameters including number of parity-check digits (n − k), minimum distance dmin and error correcting capability t are readily exposed.

Anais da Academia Brasileira de Ciências, 2013

For a given binary BCH code Cn of length n = 2 s - 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.

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.

In the present Digital Communication systems, it is highly possible that the data or message get corrupted during transmission and reception through a noisy channel medium. To get the error free communication we need Error correction code. BCH codes invented in 1960s are powerful class of multiple error correction codes with well defined mathematical properties, used to correct multiple random error patterns. The mathematical properties within which BCH codes are defined are the Galois Field or Finite Field Theory. The project proposed is " FPGA implementation of Encoder and decoder for (15, 11, 3) and (63, 39, 4) Binary BCH code using VHDL with multiple error correction ". The digital logic implementation of binary encoding and decoding of multiple error correcting BCH code of length n=15 and n=63 over GF (2 4) and GF(2 6)with irreducible primitive polynomials x 4 +x+1 and x 6 +x+1 are organized into n-k linear feedback shift register circuits for encoding. Iterative decoding algorithms are used to find the location of error and decode the message bits at receiver side. Two encoders and decoders are designed using VHDL to encode and decode the triple and four error correcting BCH code corresponding to the coefficient of generated polynomial. For implementation Spartan 3 FPGA processor is used with VHDL and the simulation & synthesis are performed using Xilinx ISE 13.2.

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].