Self-checking circuits and decoding algorithms for binary hamming and BCH codes and Reed-Solomon codes over GF(2 m )

Abstract

We consider problems of detecting errors in combinational circuits and algorithms for the decoding of linear codes. We show that a totally self-checking combinatorial circuit for the decoding of a binary Hamming [n, k] code can be constructed if and only if n = 2 r ? 1, r = n?k. We introduce the notion of a totally self-checking combinational circuit detecting error clusters of size at most µ; for shortened Hamming [n,k] codes, we construct totally self-checking decoding combinational circuits detecting error clusters of size at most µ, 2 ? µ < n?k. We describe single-error protected and self-checking algorithms: the extended Euclidean algorithm and decoding algorithms for binary BCH codes and Reed-Solomon codes over GF(2 m ).