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In this paper, the concept of the {\it broken diagonal pair} in thechess-like square board is used to define some well-structured block designswhose incidence matrices can be considered as the parity-check matrices of somehigh rate cycle codes with girth 12. The structure of the proposed parity-checkmatrices significantly reduces the complexity of encoding and decoding.Interestingly, the constructed regular cycle codes with row-weights $t$, $3\leqt \leq 20$, $t\neq 7, 15, 16$, have the best lengths among the known regulargirth-12 cycle codes. In addition, the proposed cycle codes can be easilyextended to some high rate column weight-3 LDPC codes with girth 6. Simulationresults show that the constructed codes achieve excellent performances,specially the constructed column weight 3 LDPC codes outperform LDPC codesbased on Steiner triple systems (STS).

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