Deterministic Girth-Eight QC-LDPC Codes with Large Column Weight

For any row weight L, several novel classes of (J = 5, L) and (J = 6, L) quasi-cyclic LDPC codes are deterministically constructed with girth eight. From these results, it is proved that (5, L) QC-LDPC codes with girth eight exist for any circulant permutation matrix (CPM) size P ≥ (2L + 3)(L - 1) + 1, and that girth-eight (6, L) QC-LDPC codes exist for any P ≥ 2(L+5)(L-1)+1. The two novel bounds remarkably improve the existing bounds of L²(L-1) + 1 and (L²+1)(L-1)+1, respectively. Moreover, for any column weight J and any row weight L, a construction for (J,L) QC-LDPC codes with girth eight is also proposed. This is the first deterministic and systematic construction which can generate girth-eight QC-LDPC codes with J ≥ 7.