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638, 644 ] ] }, "distance": { "d": 6, "X": { "value": 6, "confidence": "exact", "witness": [ 0, 3, 8, 10, 27, 233 ] }, "Z": { "value": 6, "confidence": "exact", "witness": [ 0, 1, 3, 4, 11, 67 ] } }, "provenance": { "authors": [ "@TimoKellerMath" ], "origin": "submission", "novelty": "new_parameters", "construction": "Surface code of the orientably-regular map C163.13 of type {8,8} (genus 163) in M. Conder's list of chiral maps of genus 2 to 301. Rotation group G = of order 1296. Darts = elements of G (right multiplication), faces = cosets g, vertices = cosets g, edges = cosets g = {g, grs}. Qubits on the 648 edges, X-checks on the 162 vertices (weight 8), Z-checks on the 162 faces (weight 8), incidences mod 2: H_X and H_Z are the boundary maps of the cellular chain complex over F_2, k = 2 - chi = 326, d_Z is the shortest homologically nontrivial cycle of the map and d_X that of the dual map. Found in a census of regular maps (normal subgroups of the triangle groups , Magma's LowIndexNormalSubgroups). The qubit order is that of the dart permutations in the public package in the references (data/maps.json), whose src/rebuild.py rebuilds these matrices from them and from the presentation.", "references": [ "https://arxiv.org/abs/1506.04029", "https://www.math.auckland.ac.nz/~conder/ChiralMaps301.txt", "https://github.com/TimoKellerMath/quantum-error-correction/tree/2e97a5361e7b9327128baa242bdedd333f7d7e12/hyperbolic-map-codes" ], "date": "2026-11-10", "model": "Claude Opus 5.5", "notes": "Exact distance d_X = d_Z = 6: tropidist (arXiv:2611.08247) at the board's pinned commit b03ec2d4077f closes both sides (certs/648-326-6.json, level solver); independently two exact programs for check matrices with column weight 2 and an enumeration of all circuits of length <= 6 (24732 X-type and 24732 Z-type logical operators of weight 6, none lighter), see the research note. Novelty: no board entry with these n and k at commit 8042f9da; parameters not found in Breuckmann and Terhal (arXiv:1506.04029), Breuckmann et al. (arXiv:1703.00590), Breuckmann's thesis (arXiv:1802.01520), Mahmoud and Rayan (arXiv:2610.10948), Breuckmann and Burton (arXiv:2202.06647), Mahmoud, Ali and Rayan (arXiv:2504.07800), the catalogs of QEC-pages (Quantum_LDPC_Codes) and qecdb.org; Conder's lists contain the map, not the code. Equivalence with existing entries: checked, none has these n and k." }, "family": "topological" }