A Separation Algorithm for Improved LP-Decoding of Linear Block Codes

TL;DR

This paper introduces a new integer programming formulation and a separation algorithm with Gomory cuts for improved maximum likelihood decoding of linear block codes, outperforming traditional LP decoding.

Contribution

The authors propose a novel IP formulation and a separation algorithm using Gomory cuts, including RPC cuts, to enhance ML decoding performance.

Findings

Significantly better decoding performance than LP decoding on tested codes.

Effective use of Gomory cuts and RPC cuts to eliminate fractional solutions.

Demonstrated improvements on LDPC and BCH codes.

Abstract

Maximum Likelihood (ML) decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an Integer Programming (IP) problem. Feldman et al. relaxed this IP problem and presented Linear Programming (LP) based decoding algorithm for linear block codes. In this paper, we propose a new IP formulation of the ML decoding problem and solve the IP with generic methods. The formulation uses indicator variables to detect violated parity checks. We derive Gomory cuts from our formulation and use them in a separation algorithm to find ML codewords. We further propose an efficient method of finding cuts induced by redundant parity checks (RPC). Under certain circumstances we can guarantee that these RPC cuts are valid and cut off the fractional optimal solutions of LP decoding. We demonstrate on two LDPC codes and one BCH code that our separation algorithm performs significantly better than LP decoding.