Construction and decoding of BCH codes over chain of commutative rings
1 Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan
2 Department of Mathematics, São Paulo State University, São José do Rio Preto, São Paulo, 15054-000, Brazil
Mathematical Sciences 2012, 6:51 doi:10.1186/2251-7456-6-51Published: 12 October 2012
In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let be a chain of unitary commutative rings, where each is constructed by the direct product of appropriate Galois rings, and its projection to the fields is (another chain of unitary commutative rings), where each is made by the direct product of corresponding residue fields of given Galois rings. Also, and are the groups of units of and , respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from and 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.