10.1186/2251-7456-6-51

Construction and decoding of BCH codes over chain of commutative rings

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  • Tariq Shah1,
  • Attiq Qamar1,
  • Antonio Aparecido de Andrade*2,
  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, PK
  2. Department of Mathematics, São Paulo State University, São José do Rio Preto, São Paulo, 15054-000, BR

Published in Issue 2012-10-12

How to Cite

Shah, T., Qamar, A., & de Andrade, A. A. (2012). Construction and decoding of BCH codes over chain of commutative rings. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-51
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Abstract

Abstract In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative 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 appropriate Galois rings, and its projection to the fields is K0⊂K1⊂⋯⊂Kt−1⊂Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, Ai∗ and Ki∗ are the groups of units of Ai and Ki , respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from Ai∗ and Ki∗ 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 error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

Keywords

  • Units of a Galois ring,
  • BCH code,
  • McCoy rank,
  • Direct product of Galois rings,
  • 11T71,
  • 94A15,
  • 14G50
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References

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