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705, 798, 810, 813, 855 ], [ 541, 636, 688, 737, 746, 747, 778, 794 ], [ 313, 461, 543, 598, 693, 717, 750, 833 ], [ 509, 554, 603, 632, 728, 747, 760, 797 ], [ 558, 731, 750, 765, 766, 835, 845, 860 ], [ 333, 460, 539, 574, 577, 703, 711, 779 ], [ 335, 438, 561, 564, 690, 771, 779, 844 ], [ 589, 666, 753, 756, 826, 837, 844, 859 ], [ 622, 630, 726, 806, 807, 811, 834, 848 ], [ 635, 655, 663, 694, 793, 804, 808, 815 ], [ 412, 525, 661, 680, 740, 809, 830, 850 ], [ 435, 685, 691, 712, 716, 780, 783, 837 ], [ 479, 503, 523, 551, 628, 665, 739, 807 ], [ 522, 648, 674, 789, 839, 847, 858, 861 ], [ 484, 668, 719, 723, 732, 749, 824, 838 ], [ 518, 676, 699, 736, 761, 828, 840, 855 ], [ 738, 821, 829, 840, 853, 856, 860, 863 ], [ 742, 817, 827, 838, 854, 857, 859, 862 ] ] }, "distance": { "d": 6, "X": { "value": 6, "confidence": "exact", "witness": [ 0, 1, 3, 10, 27, 86 ] }, "Z": { "value": 6, "confidence": "exact", "witness": [ 0, 1, 4, 7, 11, 152 ] } }, "provenance": { "authors": [ "@TimoKellerMath" ], "origin": "submission", "novelty": "new_parameters", "construction": "Surface code of the orientably-regular map C217.22 of type {8,8} (genus 217) in M. Conder's list of chiral maps of genus 2 to 301, in its mirror image (Conder's relators hold for (r^-1, s^-1); mirror images give the same code). Rotation group G = of order 1728. Darts = elements of G (right multiplication), faces = cosets g, vertices = cosets g, edges = cosets g = {g, grs}. Qubits on the 864 edges, X-checks on the 216 vertices (weight 8), Z-checks on the 216 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 = 434, 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/864-434-6.json, level solver); independently two exact programs for check matrices with column weight 2 and an enumeration of all circuits of length <= 6 (21744 X-type and 21744 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" }