# High Rate LDPC Codes from Difference Covering Arrays

## TL;DR

This paper introduces a new combinatorial method to construct high-rate LDPC codes of specific lengths using difference covering arrays, achieving desirable properties like high girth and minimum distance.

## Contribution

It presents a novel construction of LDPC codes from difference covering arrays that works for all n, overcoming limitations of previous design-based methods.

## Key findings

- Codes have length $4n^2 - 2n$ for all n.
- Achieve high information rate (>0.8) for n ≥ 8.
- Have girth at least 6 and minimum distance 6 for odd n.

## Abstract

This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from difference covering arrays. While the original construction by Gallagher was by randomly allocating bits in a sparse parity-check matrix, over the past 20 years researchers have used a variety of more structured approaches to construct these codes, with the more recent constructions of well-structured LDPC coming from balanced incomplete block designs (BIBDs) and from Latin squares over finite fields. However these constructions have suffered from the limited orders for which these designs exist. Here we present a construction of LDPC codes of length $4n^2 - 2n$ for all $n$ using the cyclic group of order $2n$. These codes achieve high information rate (greater than 0.8) for $n \geq 8$, have girth at least 6 and have minimum distance 6 for $n$ odd.