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arXiv · 2606.32001

Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding

Kenta Kasai
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Abstract

We study hard-erasure belief-propagation (BP), understood as the sum-product algorithm on erasure messages, for a punctured MacKay-Neal/Hsu-Anastasopoulos (MN/HA) representation of Calderbank-Shor-Steane (CSS) codes. For every integer degree triple $2\leq j_Z<j_X<k\leq60$, we prove that the constituent potential thresholds are $j_Z/k$ and $1-j_X/k$. The proof combines classical HA-MN duality with a completed exact rational certificate for all 1711 MN degree pairs in this range. The certificate establishes strict positivity on every physical nontrivial fixed-point branch, at every erasure probability in $[0,1]$, and has an independent integer-arithmetic verifier. A reduction of the Z-side recursion and a fixed-channel vector-potential argument then prove seeded density-evolution convergence below the smaller constituent threshold for sufficiently large coupling width and an ideal seed interval at least that wide. Under $j_Z+j_X=k$, this threshold equals the quantum-erasure hashing-bound parameter determined by the CSS design rate. In particular, the degree triple $(4,8,12)$ has potential threshold $1/3$ without a positivity hypothesis. The numerical example at erasure probability $0.3325$ reproduces the resulting decoding waves. The theorem uses an ideal auxiliary-message seed; its finite-code realization and logical block-error convergence are separate questions.

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Kenta Kasai. 2026-09-13. Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding. https://arxiv.org/abs/2606.32001

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