An Efficient Pseudo-Codeword Search Algorithm for Linear Programming Decoding of LDPC Codes

TL;DR

This paper introduces a heuristic pseudo-codeword search algorithm for LP decoding of LDPC codes, enabling efficient identification of neighboring pseudo-codewords and their effective distances to assess decoding performance.

Contribution

The authors propose a novel iterative algorithm to find and analyze pseudo-codewords near the zero codeword in LP decoding, improving understanding of decoding errors at various SNR levels.

Findings

Algorithm effectively finds neighboring pseudo-codewords.

Distribution of pseudo-codeword distances characterized.

Approximate effective distance computed for specific LDPC codes.

Abstract

In Linear Programming (LP) decoding of a Low-Density-Parity-Check (LDPC) code one minimizes a linear functional, with coefficients related to log-likelihood ratios, over a relaxation of the polytope spanned by the codewords \cite{03FWK}. In order to quantify LP decoding, and thus to describe performance of the error-correction scheme at moderate and large Signal-to-Noise-Ratios (SNR), it is important to study the relaxed polytope to understand better its vertexes, so-called pseudo-codewords, especially those which are neighbors of the zero codeword. In this manuscript we propose a technique to heuristically create a list of these neighbors and their distances. Our pseudo-codeword-search algorithm starts by randomly choosing the initial configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two sub-steps. Firstly, one applies an LP decoder to the noise-configuration deriving a pseudo-codeword. Secondly, one finds configuration of the noise equidistant from the pseudo codeword and the zero codeword. The resulting noise configuration is used as an entry for the next step. The iterations converge rapidly to a pseudo-codeword neighboring the zero codeword. Repeated many times, this procedure is characterized by the distribution function (frequency spectrum) of the pseudo-codeword effective distance. The effective distance of the coding scheme is approximated by the shortest distance pseudo-codeword in the spectrum. The efficiency of the procedure is demonstrated on examples of the Tanner $[155,64,20]$ code and Margulis $p=7$ and $p=11$ codes (672 and 2640 bits long respectively) operating over an Additive-White-Gaussian-Noise (AWGN) channel.