QC-LDPC Codes from Difference Matrices and Difference Covering Arrays

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.