Rational Maps and Maximum Likelihood Decodings

TL;DR

This paper introduces an approximate maximum likelihood decoding method for error-correcting codes using rational maps and Taylor expansion, showing improved performance over BCH codes through numerical analysis.

Contribution

It proposes a novel approximate ML decoding rule based on Taylor expansion and dynamical system properties, linking the expansion order to the dual code's minimum distance.

Findings

The order of nonlinear terms relates to the dual code's minimum distance.

Numerical results show better bit error rates than BCH codes.

The method closely approximates original ML decoding.

Abstract

This paper studies maximum likelihood(ML) decoding in error-correcting codes as rational maps and proposes an approximate ML decoding rule by using a Taylor expansion. The point for the Taylor expansion, which will be denoted by $p$ in the paper, is properly chosen by considering some dynamical system properties. We have two results about this approximate ML decoding. The first result proves that the order of the first nonlinear terms in the Taylor expansion is determined 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 better performance than that of BCH codes, and indicate that this proposed method approximates the original ML decoding very well.