Pseudo-Codeword Analysis of Tanner Graphs from Projective and Euclidean Planes

TL;DR

This paper analyzes the structure and pseudo-weights of minimal pseudo-codewords in LDPC codes derived from projective and Euclidean planes to understand their performance under LP decoding.

Contribution

It provides a detailed study of minimal pseudo-codewords in specific LDPC codes from geometric constructions and establishes lower bounds on their pseudo-weights.

Findings

Identifies structural properties of minimal pseudo-codewords.

Provides lower bounds on pseudo-weights for these codes.

Enhances understanding of LP decoding performance for geometric LDPC codes.

Abstract

In order to understand the performance of a code under maximum-likelihood (ML) decoding, one studies the codewords, in particular the minimal codewords, and their Hamming weights. In the context of linear programming (LP) decoding, one's attention needs to be shifted to the pseudo-codewords, in particular to the minimal pseudo-codewords, and their pseudo-weights. In this paper we investigate some families of codes that have good properties under LP decoding, namely certain families of low-density parity-check (LDPC) codes that are derived from projective and Euclidean planes: we study the structure of their minimal pseudo-codewords and give lower bounds on their pseudo-weight.