QC-LDPC Codes from Difference Matrices and Difference Covering Arrays

TL;DR

This paper introduces a new framework for constructing LDPC codes using difference matrices and covering arrays, enabling broader code lengths and rates, with proven properties and competitive performance.

Contribution

It presents a novel LDPC code construction method based on difference matrices, extending applicability beyond finite field-based designs, with verified minimum and stopping distances.

Findings

Codes have length a^2 or a^2 - a depending on parity of a.

Constructed codes achieve minimum distance at least 8 or 10 under certain conditions.

Performance over AWGN is comparable or superior to existing codes.

Abstract

We give a framework for generalizing LDPC code constructions that use Transversal Designs or related structures such as mutually orthogonal Latin squares. Our construction offers a broader range of code lengths and codes rates. Similar earlier constructions rely on the existence of finite fields of order a power of a prime. In contrast the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures available for any order $a$. They satisfy the RC constraint and have, for $a$ odd, length $a^2$ and rate $1-\frac{4a-3}{a^2}$, and for $a$ even, length $a^2-a$ and rate at least $1-\frac{4a-6}{a^2-a}$. When $3$ does not divide $a$, these LDPC codes have stopping distance at least $8$. When $a$ is odd and both $3$ and $5$ do not divide $a$, our construction delivers an infinite family of QC-LDPC codes with minimum distance at least $10$. The simplicity of the construction allows us to theoretically verify these properties and analytically determine lower bounds for the minimum distance and stopping distance of the code. The BER and FER performance of our codes over AWGN (via simulation) is at the least equivalent to codes constructed previously, while in some cases significantly outperforming them.