High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes

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 constacyctic maximum distance separable code of dimension 2. The construction is similar to the derivation of Reed-Solomon (RS)-based LDPC codes presented by Chen et al. and Djurdjevic et al. Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes.