Sequences of Primitive and Non-primitive BCH Codes

A. S. Ansari, T. Shah, Zia Ur-Rahman, Antonio A. Andrade

Abstract


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 $\{C_{b^{j}n}\}_{1\leq j\leq m}$, 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 built 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.

Keywords


Monoid ring, BCH codes; primitive polynomial; non-primitive polynomial.

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References


A. V. Kelarev, and P. Sole, Error-correcting codes as ideals in group ring, Contemporary Mathematics, 273, 11-18, (2001).

J. Cazaran, A.V. Kelarev, S.J. Quinn, D. Vertigan, An algorithm for computing the minimum distances of extensions of BCH codes embedded in semigroup rings, Semigroup Forum, 73, 317-329, (2006).

A. A. Andrade and Palazzo Jr., Linear codes over finite rings, TEMA-Tend. Mat. Apl. Comput., 6(2), 207-217, 2005.




DOI: https://doi.org/10.5540/tema.2018.019.02.369

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TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
ISSN: 1677-1966  (print version),  2179-8451  (online version)

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