Graph based linear error correcting codes

TL;DR

This paper introduces a new method for constructing Low Density Parity Check (LDPC) codes using bipartite, bi-regular graphs without short cycles, resulting in fast-decodable error-correcting codes close to Shannon bounds.

Contribution

The paper presents a novel construction of LDPC codes based on special graphs by Ustimenko and Woldar, emphasizing their sparse, cycle-free structure for improved decoding performance.

Findings

Codes exhibit low bit error rates in simulations.

Construction allows for adjustable code rates.

Codes are fast to decode due to sparse graph structure.

Abstract

In this article we present a construction of error correcting codes, that have representation as very sparse matrices and belong to the class of Low Density Parity Check Codes. LDPC codes are in the classical Hamming metric. They are very close to well known Shannon bound. The ability to use graphs for code construction was first discussed by Tanner in 1981 and has been used in a number of very effective implementations. We describe how to construct such codes by using special a family of graphs introduced by Ustimenko and Woldar. Graphs that we used are bipartite, bi-regular, very sparse and do not have short cycles C 4 . Due to the very low density of such graphs, the obtained codes are fast decodable. We describe how to choose parameters to obtain a desired code rate. We also show results of computer simulations of BER (bit error rate) of the obtained codes in order to compare them with other known LDPC codes.