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A method to construct girth-12 (3,L) quasi-cyclic low-density parity-check(QC-LDPC) codes with all lengths larger than a certain given number isproposed, via a given girth-12 code subjected to some constraints. The lengthsof these codes can be arbitrary integers of the form LP, provided that P islarger than a tight lower bound determined by the maximal element within theexponent matrix of the given girth-12 code. By applying the method to the caseof row-weight six, we obtained a family of girth-12 (3,6) QC-LDPC codes forarbitrary lengths above 2688, which includes 30 member codes with shorter codelengths compared with the shortest girth-12 (3,6) QC-LDPC codes reported byO'Sullivan.

Girth-12 Quasi-Cyclic LDPC Codes with Consecutive Lengths