Column Weight Two and Three LDPC Codes with High Rates and Large Girths

In this paper, the concept of the {\it 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. The structure of the proposed parity-check matrices significantly reduces the complexity of encoding and decoding. Interestingly, the constructed regular cycle codes with row-weights $t$, $3\leq t \leq 20$, $t\neq 7, 15, 16$, have the best 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 codes achieve excellent performances, specially the constructed column weight 3 LDPC codes outperform LDPC codes based on Steiner triple systems (STS).