Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction

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.

arXiv commentv2 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