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For an arbitrary (3,L) quasi-cyclic(QC) low-density parity-check (LDPC) code with girth at least ten, a tight lower bound of the consecutive lengths is presented. For an arbitrary length above the bound the corresponding LDPC code necessarily has a girth at least ten, and for the length equal to the bound, the resultant code inevitably has a girth smaller than ten. This new conclusion can be well applied to some important issues, such as the proofs of the existence of large girth QC-LDPC codes, the construction of large girth QC-LDPC codes based on the Chinese remainder theorem, as well as the construction of LDPC codes with the guaranteed error correction capability.

Tight lower bound of consecutive lengths for QC-LDPC codes with girth at least ten