Tight lower bound of consecutive lengths for QC-LDPC codes with girth twelve

Abstract: For an arbitrary (3,L) QC-LDPC code with a girth of twelve, a tight lower bound of the consecutive lengths is proposed. For an arbitrary length above the bound the resultant code necessarily has a girth of twelve, and for the length meeting the bound, the corresponding code inevitably has a girth smaller than twelve. The conclusion can play an important role in 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, and the construction of LDPC codes with the guaranteed error correction capability.