{
 "schema_version": "0.2",
 "name": "[[37,7,3]] punctured triangular colour-code complex",
 "code_type": "CSS",
 "n": 37,
 "k": 7,
 "checks": {
  "X": [
   [
    1,
    2,
    7,
    8
   ],
   [
    5,
    6,
    11,
    12
   ],
   [
    0,
    1,
    7,
    13
   ],
   [
    2,
    3,
    8,
    9,
    14,
    15
   ],
   [
    4,
    5,
    10,
    11,
    16,
    17
   ],
   [
    7,
    8,
    13,
    14,
    18,
    19
   ],
   [
    9,
    10,
    15,
    16,
    20,
    21
   ],
   [
    11,
    12,
    17,
    22
   ],
   [
    14,
    15,
    19,
    20,
    23,
    24
   ],
   [
    16,
    17,
    21,
    22,
    25,
    26
   ],
   [
    20,
    21,
    24,
    25,
    28,
    29
   ],
   [
    23,
    24,
    27,
    28,
    30,
    31
   ],
   [
    28,
    29,
    31,
    32,
    33,
    34
   ],
   [
    30,
    31,
    33,
    35
   ],
   [
    33,
    34,
    35,
    36
   ]
  ],
  "Z": [
   [
    1,
    2,
    7,
    8
   ],
   [
    5,
    6,
    11,
    12
   ],
   [
    0,
    1,
    7,
    13
   ],
   [
    2,
    3,
    8,
    9,
    14,
    15
   ],
   [
    4,
    5,
    10,
    11,
    16,
    17
   ],
   [
    7,
    8,
    13,
    14,
    18,
    19
   ],
   [
    9,
    10,
    15,
    16,
    20,
    21
   ],
   [
    11,
    12,
    17,
    22
   ],
   [
    14,
    15,
    19,
    20,
    23,
    24
   ],
   [
    16,
    17,
    21,
    22,
    25,
    26
   ],
   [
    20,
    21,
    24,
    25,
    28,
    29
   ],
   [
    23,
    24,
    27,
    28,
    30,
    31
   ],
   [
    28,
    29,
    31,
    32,
    33,
    34
   ],
   [
    30,
    31,
    33,
    35
   ],
   [
    33,
    34,
    35,
    36
   ]
  ]
 },
 "distance": {
  "d": 3,
  "X": {
   "value": 3,
   "confidence": "upper_bound",
   "witness": [
    32,
    33,
    35
   ]
  },
  "Z": {
   "value": 3,
   "confidence": "upper_bound",
   "witness": [
    4,
    5,
    6
   ]
  }
 },
 "provenance": {
  "authors": [
   "@mathysrennela"
  ],
  "origin": "submission",
  "novelty": "unknown",
  "model": "OpenAI GPT-5.6 Luna",
  "construction": "Punctured self-dual CSS subcomplex obtained from the [[37,1,7]] triangular 6.6.6 colour-code face-incidence matrix by removing face checks 1, 11, and 14; preserves the parent single-layer non-affine layout.",
  "references": [
   "arXiv:quant-ph/0605138",
   "arXiv:1108.5738",
   "errorcorrectionzoo.org/c/triangular_color"
  ],
  "date": "2026-08-02",
  "notes": "Research candidate promoted for human review. Exact MILP certification with verify/certify.py --tlim 120 proved d_X=d_Z=3; the submitted distance confidence remains upper_bound per the challenge workflow. Operational efficiency is 1.7027027027 and geometric efficiency is 0.4889746561."
 },
 "family": "topological",
 "locality": {
  "coordinates": [
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    4.203408830010462,
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  ],
  "interaction_radius": 1.9318709710946886,
  "layers": 1
 },
 "circuit": {
  "d_circ": {
   "X": {
    "value": 3,
    "confidence": "upper_bound",
    "witness": [
     16,
     25,
     53
    ]
   },
   "Z": {
    "value": 3,
    "confidence": "upper_bound",
    "witness": [
     706,
     756,
     768
    ]
   }
  },
  "rounds": 3,
  "stim_version": "1.16.0",
  "notes": "Flag-qubit sequential two-block extraction: one ancilla and one flag qubit per check, X checks then Z checks each round with joint reset/measure layers. Each block has 6 data CNOT slot layers (a proper edge coloring of the check type's Tanner graph); the flag (|0> for X checks, |+> for Z checks) is coupled to the ancilla after slot 1 and again after slot 5, and measured every round as a single-record detector, so an ancilla hook in that window also fires the flag. 8 X + 8 Z CX layers per round, schedule seed 0. Hook-aware orders: every reachable suffix S of a check's CNOT order needs 1 + [flagged] + cost(S) >= d faults and every same-ancilla pair of hooks 2 + [statuses differ] + cost >= d, where cost is the exact fewest further data errors completing the set to a logical; layered by min-conflicts search. Pairs of hooks on different ancillas of the same type are also constrained: an assignment pair is admitted only if every such hook pair needs >= d faults in total (2 + flags + completion cost). Screened by RIS on ker(H_dem) and an observable-constrained seeded BP+OSD search; witnesses validated in GF(2).",
  "contributed_by": {
   "by": [
    "@vprusso"
   ],
   "date": "2026-09-19",
   "method": "generic CSS memory schedule with one flag qubit per check: CX layers from a bipartite edge coloring, the flag coupled to the ancilla after the first and the second-to-last CX slot and measured every round, slot assignments filtered by an exact hook-cost criterion over single, same-ancilla and cross-ancilla hook pairs; screened by RIS and a seeded BP+OSD search"
  },
  "ler": {
   "X": {
    "p": 0.001,
    "shots": 10000,
    "failures": 601,
    "seed": 7,
    "decoder": "bposd-cs-10",
    "ler_per_round": 0.020894316,
    "ci95": [
     0.019269222,
     0.022654034
    ]
   },
   "Z": {
    "p": 0.001,
    "shots": 10000,
    "failures": 566,
    "seed": 7,
    "decoder": "bposd-cs-10",
    "ler_per_round": 0.019627026,
    "ci95": [
     0.01805725,
     0.02133095
    ]
   }
  }
 }
}