Bit-error probability for maximum-likelihood decoding of linear block codes and related soft-decision decoding methods

In this correspondence, the bit-error probability P_b for maximum-likelihood decoding of binary linear block codes is investigated. The contribution P_b(j) of each information bit j to P_b is considered and an upper bound on P_b(j) is derived. For randomly generated codes, it is shown that the conventional approximation at high SNR P_b≈(d_H/N).P_s, where P_s represents the block error probability, holds for systematic encoding only. Also systematic encoding provides the minimum P_b when the inverse mapping corresponding to the generator matrix of the code is used to retrieve the information sequence. The bit-error performances corresponding to other generator matrix forms are also evaluated. Although derived for codes with a generator matrix randomly generated, these results are shown to provide good approximations for codes used in practice. Finally, for soft-decision decoding methods which require a generator matrix with a particular structure such as trellis decoding, multistage decoding, or algebraic-based soft-decision decoding, equivalent schemes that reduce the bit-error probability are discussed. Although the gains achieved at practical bit-error rates are only a fraction of a decibel, they remain meaningful as they are of the same orders as the error performance differences between optimum and suboptimum decodings. Most importantly, these gains are free as they are achieved with no or little additional circuitry which is transparent to the conventional implementation