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

TL;DR

This paper establishes a precise lower bound on the consecutive lengths of (3,L) QC-LDPC codes with girth twelve, aiding in their construction and analysis.

Contribution

It introduces a tight lower bound for the lengths of (3,L) QC-LDPC codes with girth twelve, clarifying the relationship between code length and girth.

Findings

Codes above the bound have girth twelve

Codes at the bound have girth less than twelve

The bound assists in code construction and analysis

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.