Quasi-cyclic low-density parity-check codes based on finite set systems

A finite set system (FSS) is a pair (V, B) where V is a finite set whose members are called points, equipped with a finite collection of its subsets B whose members are called blocks. In this paper, FSSs are used to define a class of quasi-cyclic low-density parity-check (LDPC) codes, called FSS codes, such that the constructed codes possess large girth and arbitrary column-weight distributions. Especially, the constructed column weight-2 FSS codes have higher rates than the column weight-2 geometric and cylinder-type codes with the same girths. To find the maximum girth of FSS codes based on (V, B), inevitable walks are defined in B such that the maximum girth is determined by the smallest length of the inevitable walks in B. Simulation results show that the constructed FSS codes have very good performance over the additive white Gaussian noise channel with iterative decoding and achieve significantly large coding gains compared with the random-like LDPC codes of the same lengths and rates.