Quasi-Cyclic LDPC Codes With Girth at Least Eight Based on Disjoint Difference Sets

A novel relationship between disjoint difference sets (DDS) and QC-LDPC codes without cycles of length up to six is established. Based on DDS with two sets, a class of column-weight-four QC-LDPC codes is constructed with girth at least eight. In the literature there are several explicit or semi-random methods which can yield column-weight-four QC-LDPC codes without 4-cycles and 6-cycles. Compared with these existing methods, the new DDS-based method is able to offer QC-LDPC codes with desirable decoding performance and flexible circulant sizes, and at the same time able to provide a large number of non-equivalent candidate codes. Moreover, by employing DDS with only one set, a class of column-weight-three QC-LDPC codes is constructed with short lengths and with girth at least eight.