A Branch-and-Cut Algorithm to Design LDPC Codes without Small Cycles in Communication Systems

TL;DR

This paper presents a branch-and-cut algorithm for designing LDPC codes that avoid small cycles in their Tanner graph, improving error correction performance in communication systems.

Contribution

It introduces an integer programming formulation and a novel branch-and-cut algorithm to generate LDPC codes free of small cycles, with effective heuristics and structural analysis.

Findings

Algorithm efficiently generates codes without small cycles

Codes improve error correction in noisy channels

Method applicable to practically relevant code lengths

Abstract

In a digital communication system, information is sent from one place to another over a noisy communication channel using binary symbols (bits). Original information is encoded by adding redundant bits, which are then used by low--density parity--check (LDPC) codes to detect and correct errors that may have been introduced during transmission. Error correction capability of an LDPC code is severely degraded due to harmful structures such as small cycles in its bipartite graph representation known as Tanner graph (TG). We introduce an integer programming formulation to generate a TG for a given smallest cycle length. We propose a branch-and-cut algorithm for its solution and investigate structural properties of the problem to derive valid inequalities and variable fixing rules. We introduce a heuristic to obtain feasible solutions of the problem. Our computational experiments show that our algorithm can generate LDPC codes without small cycles in acceptable amount of time for practically relevant code lengths.