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

We propose a method for constructing quantum error-correcting codes using non-binary low-density parity-check codes whose Tanner graphs have girth 16. While conventional constructions based on circulant permutation matrices are limited to a maximum girth of 12, our approach leverages affine permutation matrices combined with a randomized sequential selection strategy 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 of the proposed codes and found that they achieve a larger upper bound than conventional constructions.