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

In this paper we are interested in MAP decoding of first order Reed-Muller and Hamming codes. A MAP decoder of a linear code computes a-posteriori probabilities for all values of each transmitted symbol and chooses the maximum one. The standard way of doing this is the BCJR algorithm. This algorithm is based on a trellis representation of a code. Such a representation allows one to reduce the complexity of MAP decoding, though it still remains exponential. The BCJR algorithm for binary first order Reed-Muller and Hamming codes has complexity proportional to n², where n is the code length. In this paper we propose a new MAP decoding algorithm for both RM-1 and Hamming codes with complexity proportional to nlog₂n.