Rational maps and maximum likelihood decodings

This paper studies maximum likelihood (ML) decoding in error-correcting codes as a rational map and proposes a new approximate ML decoding rule by using the Taylor expansion. The point for the Taylor expansion, which will be denoted by p in the paper, is properly chosen by considering some properties of the rational map dynamical systems. We have two results about this approximate ML decoding. The first result proves that the first nonlinear terms in the Taylor expansion are determined by linear dependent column vectors of a generator matrix. In particular, the order of the first nonlinearity is given by the minimum distance of its dual code. As the second result, we give numerical results on bit error probabilities for the approximate ML decoding. These numerical results show performances better than those of BCH codes, and indicate that this proposed method approximates the original ML decoding very well.