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  ]
 },
 "distance": {
  "d": 6,
  "X": {
   "value": 6,
   "confidence": "exact",
   "witness": [
    0,
    3,
    10,
    11,
    26,
    59
   ]
  },
  "Z": {
   "value": 6,
   "confidence": "exact",
   "witness": [
    0,
    1,
    4,
    19,
    42,
    59
   ]
  }
 },
 "provenance": {
  "authors": [
   "@TimoKellerMath"
  ],
  "origin": "submission",
  "novelty": "new_parameters",
  "construction": "Surface code of the orientably-regular map R82.22 of type {6,6} (genus 82) in M. Conder's list of reflexible maps of genus 2 to 301. Rotation group G = <r,s | r^6, s^6, (r*s)^2, s*r*s^-2*r^3*s^3*r^-2, r*s^-2*r*s^-1*r^2*s^-2*r*s^-1*r, s*r*s^-1*r^-1*s*r^2*s*r^-1*s^-1*r*s> of order 972. Darts = elements of G (right multiplication), faces = cosets g<r>, vertices = cosets g<s>, edges = cosets g<rs> = {g, grs}. Qubits on the 486 edges, X-checks on the 162 vertices (weight 6), Z-checks on the 162 faces (weight 6), incidences mod 2: H_X and H_Z are the boundary maps of the cellular chain complex over F_2, k = 2 - chi = 164, 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 <r,s | r^p, s^q, (rs)^2>, 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/RegularOrientableMaps301.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/486-164-6.json, level solver); independently two exact programs for check matrices with column weight 2 and an enumeration of all circuits of length <= 6 (2835 X-type and 2835 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"
}