On the Joint Decoding of LDPC Codes and Finite-State Channels via Linear Programming

TL;DR

This paper extends linear programming decoding to joint decoding of binary linear codes over finite-state channels, providing theoretical bounds and insights into decoder performance and error probabilities.

Contribution

It introduces a rigorous framework for LP joint-decoding pseudo-codewords and analyzes their impact on decoding performance for finite-state channels.

Findings

Provides a bound on decoder failure probability.

Shows LP joint decoder has ML certificate property for finite-state channels.

Explains performance loss via fractional pseudo-codewords.

Abstract

In this paper, the linear programming (LP) decoder for binary linear codes, introduced by Feldman, et al. is extended to joint-decoding of binary-input finite-state channels. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the LP joint decoder also has the maximum-likelihood (ML) certificate property and all integer valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional valued JD-PCWs.