A Number Theoretic Approach to Cycles in LDPC Codes

TL;DR

This paper introduces a number-theoretic framework using Sidon sets to construct LDPC codes with large girth, aiming to reduce short cycles in their Tanner graphs, and extends the approach to irregular codes.

Contribution

It presents a novel number-theoretic method for constructing LDPC codes with specified girth, connecting cycle criteria to Sidon sets, and extends the approach to irregular LDPC codes.

Findings

Constructed LDPC codes with guaranteed girth using number theory.

Established a link between cycle length criteria and Sidon sets.

Extended the construction method to irregular LDPC codes.

Abstract

LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanner graph, i.e. obtaining a possibly large girth. In this paper, we provide a framework to obtain constructions of such codes. We relate criteria for the existence of cycles of a certain length with some number-theoretic concepts, in particular with the so-called Sidon sets. In this way we obtain examples of LDPC codes with a certain girth. Finally, we extend our constructions to also obtain irregular LDPC codes.