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arXiv · 2604.20838

High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures

Koki Okada·Kenta Kasai
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Abstract

We construct a quantum low-density parity-check code family from a length-$512$ Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are $(3,8)$-regular with girth $8$. The base code has parameters $[[512,174,8]]$. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor $P=32$, which has parameters $[[16384,4142,18\le d\le32]]$, under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most $16$ on both sides, giving $d_X,d_Z\ge18$. Explicit weight-$32$ non-stabilizer logical representatives give the upper bounds $d_X,d_Z\le32$. A belief-propagation decoder with post-processing achieved frame error rate about $10^{-8}$ at $p=0.085$; an independently observed logical residual of weight $40$ is consistent with the sharper structural bound.

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Koki Okada, Kenta Kasai. 2026-09-11. High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures. https://arxiv.org/abs/2604.20838

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