Constructions and decoding of a sequence of BCH codes

Abstract

The BCH code C (respectively, C) of length n over a local ring Z pk (respectively, ℤp) is an ideal in the ring (Equation Presented) (respectively, (Equation Presented) which is generated by a monic polynomial that divides Xn - 1. Shankar [12] has shown that the roots of Xn - 1 are the unit elements of a suitable Galois ring extension GR(pk,s) (respectively, Galois field extension GF(p, s)) of the ring ℤpk (respectively, ℤp), where s is the degree of basic irreducible polynomial f(X) ∈ ℤpk [X]. In this study we assume that for st = bi, where 6 is prime and t is a non negative integer such that 0 ≤ i ≤ t, there exist corresponding chain of Galois ring extensions GR(pk, s,) (respectively, a chain of Galois field extensions GF(p, s,)) of ℤpk (respectively, ℤp), there are two situations; st = bi for i = 2 or st = bi for i ≥ 2. Consequently, the case is alike [12] and we obtain a sequence of BCH codes C0,C1, ···, Ct-1, C over ℤpk and C′0,C′1,···, C′t-1,C′ over ℤp with lengths n 0,n1,···, nt-1,n t. In second phase we extend the Modified Berlekamp-Massey Algorithm for the chain of Galois rings in such a way that the error will be corrected of the sequence of codewords from the sequence of BCH codes C0,C 1, ···, Ct-1,C. © Global Publishing Company.