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This research introduces a novel construction method using affine permutation matrices to achieve girth-16 in non-binary LDPC codes, surpassing the conventional limit of 12. The study demonstrates that these enhanced codes significantly reduce low-weight codewords and improve error floor performance under joint belief propagation decoding without sacrificing waterfall region efficiency. Such advancements are critical for scaling fault-tolerant quantum computing architectures by enabling more robust error correction schemes.
Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction
arXiv:2504.17790v3 Announce Type: replace Abstract: We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with Tanner graph girth 16. While conventional constructions using circulant permutation matrices are limited to girth 12, our method employs affine permutation matrices and a randomized sequential selection procedure to eliminate short cycles and achieve girth 16. Numerical experiments show that the proposed codes significantly reduce the number of low-weight codewords. Joint belief propagation decoding over depolarizing channels reveals that although a slight degradation appears in the waterfall region, a substantial improvement is achieved in the error floor performance. We also evaluated the minimum distance and found that the proposed codes achieve a larger upper bound compared to conventional constructions.
Abstract:We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with Tanner graph girth 16. While conventional constructions using circulant permutation matrices are limited to girth 12, our method employs affine permutation matrices and a randomized sequential selection procedure to eliminate short cycles and achieve girth 16. Numerical experiments show that the proposed codes significantly reduce the number of low-weight codewords. Joint belief propagation decoding over depolarizing channels reveals that although a slight degradation appears in the waterfall region, a substantial improvement is achieved in the error floor performance. We also evaluated the minimum distance and found that the proposed codes achieve a larger upper bound compared to conventional constructions.
v2 corrected an error-floor detection bug and updated the numerical results. v3: Corrected the formal H_Z definition by swapping the f^{-1}/g^{-1} halves to match the example and orthogonality proof; repaired the linked code archive so unsuffixed Gamma is a plain-text adjacency file
Quantum Physics (quant-ph); Information Theory (cs.IT)
arXiv:2504.17790 [quant-ph]
(or arXiv:2504.17790v3 [quant-ph] for this version)
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