{
 "schema_version": "0.1",
 "name": "[[31,21,3]]",
 "code_type": "CSS",
 "n": 31,
 "k": 21,
 "checks": {
  "X": [
   [
    0,
    2,
    4,
    6,
    8,
    10,
    12,
    14,
    16,
    18,
    20,
    22,
    24,
    26,
    28,
    30
   ],
   [
    1,
    2,
    5,
    6,
    9,
    10,
    13,
    14,
    17,
    18,
    21,
    22,
    25,
    26,
    29,
    30
   ],
   [
    3,
    4,
    5,
    6,
    11,
    12,
    13,
    14,
    19,
    20,
    21,
    22,
    27,
    28,
    29,
    30
   ],
   [
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    23,
    24,
    25,
    26,
    27,
    28,
    29,
    30
   ],
   [
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22,
    23,
    24,
    25,
    26,
    27,
    28,
    29,
    30
   ]
  ],
  "Z": [
   [
    0,
    2,
    4,
    6,
    8,
    10,
    12,
    14,
    16,
    18,
    20,
    22,
    24,
    26,
    28,
    30
   ],
   [
    1,
    2,
    5,
    6,
    9,
    10,
    13,
    14,
    17,
    18,
    21,
    22,
    25,
    26,
    29,
    30
   ],
   [
    3,
    4,
    5,
    6,
    11,
    12,
    13,
    14,
    19,
    20,
    21,
    22,
    27,
    28,
    29,
    30
   ],
   [
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    23,
    24,
    25,
    26,
    27,
    28,
    29,
    30
   ],
   [
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22,
    23,
    24,
    25,
    26,
    27,
    28,
    29,
    30
   ]
  ]
 },
 "distance": {
  "d": 3,
  "X": {
   "value": 3,
   "confidence": "upper_bound",
   "witness": [
    9,
    22,
    28
   ]
  },
  "Z": {
   "value": 3,
   "confidence": "upper_bound",
   "witness": [
    9,
    22,
    28
   ]
  }
 },
 "provenance": {
  "authors": [
   "@MathysRennela"
  ],
  "construction": "CSS from the Hamming [31,26,3] code and its Simplex dual: H_X = H_Z is the 5x31 matrix whose columns are all nonzero 5-bit vectors.",
  "origin": "submission",
  "date": "2026-10-20",
  "model": "Beam 1.0"
 },
 "family": "other"
}
