Searching for Voltage Graph-Based LDPC Tailbiting Codes with Large Girth

TL;DR

This paper explores methods for constructing LDPC block codes with large girth using voltage graph techniques, providing new bounds, algorithms, and specific code constructions with girth up to 24.

Contribution

It introduces novel algorithms and constructions for LDPC codes with large girth based on voltage graph principles and tailbiting, including codes with girth up to 24.

Findings

Bounds on girth and minimum distance are established.

Algorithms for searching LDPC codes with large girth are developed.

New QC regular LDPC codes with girth up to 24 are constructed.

Abstract

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-ones matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.