Linear-Congruence Constructions of Low-Density Parity-Check Codes

Quasi-cyclic LDPC codes with high girth

“…In fact in such case 6-cycle configurations will always arise independently from the positions of the diagonals. The existence of this configurations carry a parallel with the existence of 6-cycles on a weight-3 circulant matrix [33]. A graphical representation of such case is given in Figure 22.…”

“…It is well-known that any quasi-cyclic code has a parity-check matrix H made of circulant sub-matrices. However, if an LDPC code with good girth characteristics is desired, it is necessary to avoid sub-matrices that are weight-t circulant matrix, with t ≥ 3, since they contain internal 6-cycles [33] and codes with higher girth are wanted. Matrices in C m,α,β,γ are the only interesting parity-check matrices for good quasi-cyclic LDPC codes.…”

Versatile architectures for decoding a class of LDPC codes

“…Fossorier [10] used computer search to choose the shifts for the circulants and achieve girth of 6 and 8. The use of affine permutation matrices was introduced by Bond, Hui and Schmidt [11]. Methods for ensuring that the bipartite graph of H has large girth when affine permutations are used is presented in [5,12].…”