Explicit Construction of Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12

TL;DR

This paper presents an explicit algebraic method for constructing quantum QC-LDPC codes with girth 12, enhancing error correction performance and extending to non-binary and spatially-coupled codes without random search.

Contribution

An explicit algebraic construction approach for quantum QC-LDPC codes with girth 12, avoiding random search and applicable to non-binary and spatially-coupled codes.

Findings

Achieved girth 12 in quantum QC-LDPC codes

Eliminated short cycles to improve error correction

Extended construction to non-binary and spatially-coupled codes

Abstract

This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.