{
 "schema_version": "0.1",
 "name": "[[12,2,4]] three 422 blocks outer CSS composition",
 "code_type": "CSS",
 "n": 12,
 "k": 2,
 "checks": {
  "X": [
   [
    0,
    1,
    2,
    3
   ],
   [
    4,
    5,
    6,
    7
   ],
   [
    8,
    9,
    10,
    11
   ],
   [
    0,
    1,
    4,
    5,
    8,
    9
   ],
   [
    0,
    2,
    4,
    6,
    8,
    10
   ]
  ],
  "Z": [
   [
    0,
    1,
    2,
    3
   ],
   [
    4,
    5,
    6,
    7
   ],
   [
    8,
    9,
    10,
    11
   ],
   [
    0,
    2,
    4,
    5,
    9,
    10
   ],
   [
    0,
    1,
    5,
    6,
    8,
    10
   ]
  ]
 },
 "distance": {
  "d": 4,
  "X": {
   "value": 4,
   "confidence": "upper_bound",
   "witness": [
    2,
    3,
    8,
    10
   ]
  },
  "Z": {
   "value": 4,
   "confidence": "upper_bound",
   "witness": [
    4,
    5,
    8,
    9
   ]
  }
 },
 "provenance": {
  "authors": [
   "@mathysrennela"
  ],
  "construction": "Compose three [[4,2,2]] blocks, retaining X^4 and Z^4 per block. Apply a 6-qubit, 2-logical-qubit CSS outer code to the six block logical qubits. In block-logical order (X0X1, X0X2) with dual Z reps (Z0Z2, Z0Z1), use outer HX rows [1,0,1,0,1,0] and [0,1,0,1,0,1], and outer HZ rows [1,0,0,1,1,1] and [0,1,1,1,1,0]. Their restrictions to each block are invertible, preventing a single block from supporting a logical. The physical block checks are generated by multiplying the indicated inner logical representatives; all supports are stored in the submission.",
  "references": [],
  "date": "2026-09-23",
  "notes": "Checked against existing codes/12-2-4.json; no exact or Weisfeiler-Lehman equivalent. See notes/12-2-4-b.md.",
  "model": "GPT-6 Luna"
 },
 "family": "other",
 "locality": {
  "coordinates": [
   [
    0.0,
    0.0
   ],
   [
    0.0,
    1.0
   ],
   [
    1.0,
    0.0
   ],
   [
    1.0,
    1.0
   ],
   [
    2.0,
    0.0
   ],
   [
    2.0,
    1.0
   ],
   [
    3.0,
    0.0
   ],
   [
    3.0,
    1.0
   ],
   [
    4.0,
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   ],
   [
    4.0,
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   ],
   [
    5.0,
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   ],
   [
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   ]
  ],
  "interaction_radius": 5.0990195135927845,
  "layers": 1
 }
}