Bounds on the Error Probability of Nonbinary Linear Codes over the Q-ary Symmetric Channel

Abstract

The block error probability under maximum likelihood decoding of general linear block codes and code ensembles over the q-ary symmetric channel is analyzed. The analysis makes use of two types of upper bounds and it is based on the codes (code ensembles) distance spectra. The first bound generalizes Poltyrev's union bound for the binary symmetric channel. The second bound relies on typical pairs decoding. The two bounds are developed in an asymptotic setting to analyze the behavior of linear block code ensembles, in terms of error exponent and maximum-likelihood decoding threshold. A sphere-packing lower bound is also developed, to provide a reference for the finite-length performance.