Linear-Programming Decoding of Nonbinary Linear Codes

TL;DR

This paper develops a linear-programming decoding framework for nonbinary linear codes over rings, demonstrating its theoretical properties, complexity advantages, and comparable performance to traditional decoding methods in simulations.

Contribution

It introduces a novel LP decoding framework for nonbinary codes, proves key properties, and shows polynomial-time decoding with performance comparable to optimal methods.

Findings

LP decoder has maximum-likelihood certificate property

Decoding complexity can be reduced with alternative polytopes

Performance comparable to optimal hard-decision decoding in simulations

Abstract

A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the 'maximum-likelihood certificate' property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with linear-programming decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for the [11,6] ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision based decoding. LP decoding is also simulated for medium-length ternary and quaternary LDPC codes with corresponding PSK modulations over AWGN.