Construction of protograph-based LDPC codes with chordless short cycles

TL;DR

This paper demonstrates that eliminating chorded short cycles in protograph-based LDPC codes significantly increases their minimum distance, leading to improved error-correcting performance.

Contribution

It introduces necessary and sufficient conditions for constructing LDPC codes with chordless short cycles, enhancing minimum distance bounds and code performance.

Findings

Removing chorded short cycles raises minimum distance bounds.

Application to girth-6 QC-LDPC codes improves existing results.

Provides conditions for designing codes with chordless short cycles.

Abstract

Controlling small size trapping sets and short cycles can result in LDPC codes with large minimum distance $d_{\min}$. We prove that short cycles with a chord are the root of several trapping sets and eliminating these cycles increases $d_{\min}$. We show that the lower bounds on $d_{\min}$ of an LDPC code with chordless short cycles, girths 6 (and 8), and column weights $\gamma$ (and 3), respectively, are $2\gamma$ (and 10), which is a significant improvement compared to the existing bounds $\gamma+1$ (and 6). Necessary and sufficient conditions for exponent matrices of protograph-based LDPC codes with chordless short cycles are proposed for any type of protographs, single-edge and multiple-edge, regular and irregular. The application of our method to girth-6 QC-LDPC codes shows that the removal of those cycles improves previous results in the literature.