Large Girth Column-Weight Two and Three LDPC Codes

In this paper, the concept of broken diagonal pair in the chess-like square board is used to define some well-structured block designs whose incidence matrices can be considered as the parity-check matrices of some high rate cycle codes with girth 12. Interestingly, the constructed regular cycle codes with row-weights t, 3 ≤ t ≤ 20, t ≠ 7, 15, 16, have the minimum lengths among the known regular girth-12 cycle codes. In addition, the proposed cycle codes can be easily extended to some high rate column weight-3 LDPC codes with girth 6. Simulation results show that the constructed non-binary cycle codes and binary QC LDPC codes have good performances over AWGN channel.