Simple MAP decoding of first-order Reed-Muller and Hamming codes

A maximum a posteriori (MAP) probability decoder of a block code minimizes the probability of error for each transmitted symbol separately. The standard way of implementing MAP decoding of a linear code is the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm, which is based on a trellis representation of the code. The complexity of the BCJR algorithm for the first-order Reed-Muller (RM-1) codes and Hamming codes is proportional to n/sup 2/, where n is the code's length. In this correspondence, we present new MAP decoding algorithms for binary and nonbinary RM-1 and Hamming codes. The proposed algorithms have complexities proportional to q/sup 2/n log/sub q/n, where q is the alphabet size. In particular, for the binary codes this yields complexity of order n log n.