Improved linear programming decoding of LDPC codes and bounds on the minimum and fractional distance

TL;DR

This paper enhances linear programming decoding for LDPC codes by tightening relaxations, and introduces efficient algorithms for lower bounds on minimum and fractional distances, improving decoding performance and analysis.

Contribution

It proposes a new tightening method for LP relaxation, and algorithms for lower bounds on minimum and fractional distances with quadratic complexity.

Findings

Improved LP decoding performance through relaxation tightening.

Quadratic complexity algorithms for lower bounds on code distances.

Extended LP polytope characterization for generalized and nonbinary LDPC codes.

Abstract

We examine LDPC codes decoded using linear programming (LP). Four contributions to the LP framework are presented. First, a new method of tightening the LP relaxation, and thus improving the LP decoder, is proposed. Second, we present an algorithm which calculates a lower bound on the minimum distance of a specific code. This algorithm exhibits complexity which scales quadratically with the block length. Third, we propose a method to obtain a tight lower bound on the fractional distance, also with quadratic complexity, and thus less than previously-existing methods. Finally, we show how the fundamental LP polytope for generalized LDPC codes and nonbinary LDPC codes can be obtained.