On the Class of High-Rate QC-LDPC Codes With Girth 8 From Sequences Satisfied in GCD Condition

Abstract

Recently, a class of $(J,L)$ quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight is constructed based on the greatest-common-divisor (GCD) condition. For $L=3$ , an equivalence between the sequences generated by a greedy algorithm satisfying in GCD condition and a known class of integer sequences, called Stanley sequences has been proposed as an open problem. In this letter, we solve this problem in a more general case by introducing the class of 3-free sets as a generalization of Stanley sequences and showing an equivalence between 3-free sets and the sequences satisfied in GCD condition. Then, a new algorithm is proposed to find 3-free sets efficiently usually having larger size than the known methods, leading to column-weight 3 QC-LDPC codes with smaller lengths and better 8-cycle distributions. In addition, a new explicit formula is proposed to construct Stanley sequences which results in a class of girth-8 column-weight 3 QC-LDPC codes with high rates. The protograph QC-LDPC codes lifted from the constructed base matrices outperform progressive-edge-growth (PEG), random-like and some recent QC-LDPC codes with the same girth.