High-Rate Quasi-Cyclic Low-Density Parity-Check Codes Derived From Finite Affine Planes

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension <emphasis><formula><tex>$2$</tex></formula></emphasis>. The construction is similar to the derivation of Reed–Solomon (RS)-based LDPC codes presented by Chen <etal/> and Djurdjevic <etal/> Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes. </para>