On Existence of Girth-8 QC-LDPC Code with Large Column Weight: Combining Mirror-sequence with Classification Modulo Ten

TL;DR

This paper introduces algebraic methods combining mirror-sequence and classification modulo ten to construct short-length QC-LDPC codes with girth 8 and large column weights, improving bounds significantly.

Contribution

It proposes a novel algebraic construction for QC-LDPC codes with girth 8 and large column weights using mirror-sequence and row-regrouping within the GCD framework.

Findings

Improved lower bounds on circulant sizes by about 20%.

Constructed codes with smaller circulant sizes than existing benchmarks.

Applicable to arbitrary row weights for short-length codes.

Abstract

Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds.