  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 is
proposed, via a given girth-12 code subjected to some constraints. The lengths
of these codes can be arbitrary integers of the form LP, provided that P is
larger than a tight lower bound determined by the maximal element within the
exponent matrix of the given girth-12 code. By applying the method to the case
of row-weight six, we obtained a family of girth-12 (3,6) QC-LDPC codes for
arbitrary lengths above 2688, which includes 30 member codes with shorter code
lengths compared with the shortest girth-12 (3,6) QC-LDPC codes reported by
O'Sullivan.
