{
 "schema_version": "0.4",
 "name": "[[16,4,4]] weight-5 Z4-invariant non-CSS stabilizer code",
 "code_type": "stabilizer",
 "n": 16,
 "k": 4,
 "checks": {
  "S": [
   {
    "X": [
     0,
     4
    ],
    "Z": [
     1,
     3,
     4,
     11
    ]
   },
   {
    "X": [
     1,
     5
    ],
    "Z": [
     0,
     2,
     5,
     8
    ]
   },
   {
    "X": [
     2,
     6
    ],
    "Z": [
     1,
     3,
     6,
     9
    ]
   },
   {
    "X": [
     3,
     7
    ],
    "Z": [
     0,
     2,
     7,
     10
    ]
   },
   {
    "X": [
     7,
     10,
     12,
     13
    ],
    "Z": [
     10,
     12,
     14
    ]
   },
   {
    "X": [
     4,
     11,
     13,
     14
    ],
    "Z": [
     11,
     13,
     15
    ]
   },
   {
    "X": [
     5,
     8,
     14,
     15
    ],
    "Z": [
     8,
     12,
     14
    ]
   },
   {
    "X": [
     6,
     9,
     12,
     15
    ],
    "Z": [
     9,
     13,
     15
    ]
   },
   {
    "X": [
     2,
     7,
     8,
     9,
     12
    ],
    "Z": [
     2,
     12
    ]
   },
   {
    "X": [
     3,
     4,
     9,
     10,
     13
    ],
    "Z": [
     3,
     13
    ]
   },
   {
    "X": [
     0,
     5,
     10,
     11,
     14
    ],
    "Z": [
     0,
     14
    ]
   },
   {
    "X": [
     1,
     6,
     8,
     11,
     15
    ],
    "Z": [
     1,
     15
    ]
   }
  ]
 },
 "distance": {
  "d": 4,
  "P": {
   "value": 4,
   "confidence": "upper_bound",
   "witness": {
    "X": [
     0,
     1,
     2,
     5
    ],
    "Z": [
     0,
     1,
     2
    ]
   }
  }
 },
 "provenance": {
  "authors": [
   "@MicheldeMessieres"
  ],
  "construction": "Z4-invariant (quasi-cyclic) non-CSS stabilizer code: 16 qubits in 4 orbits of Z4 (qubit 4*b+g, b = orbit, g in Z4), 12 generators in 3 orbits of Z4 (each orbit = the 4 translates g -> g+t of one representative Pauli), every generator of weight <= 5. Found by a lazy SAT search over such codes (commutation encoded exactly; every nontrivial logical of weight < 4 found along the way is excluded together with its Z4 translates, then re-solved), related to the group-based (coset / Cayley-graph) constructions of non-CSS stabilizer LDPC codes of Camara, Ollivier and Tillich (arXiv:quant-ph/0502086). Orbit representatives: X0 Z1 Z3 Y4 Z11; X7 Y10 Y12 X13 Z14; Y2 X7 X8 X9 Y12. k = 16 - 12 = 4 (independent generators). Not CSS under any single-qubit Clifford.",
  "origin": "submission",
  "date": "2026-10-12",
  "model": "Claude Opus 5.5 (Claude Code)",
  "references": [
   "arXiv:quant-ph/0502086",
   "arXiv:2601.19848",
   "arXiv:2601.15446 (literature check; see the note)"
  ],
  "novelty": "unknown",
  "notes": "Distance: an exhaustive enumeration of every Pauli operator of weight 1, 2 and 3 finds no nontrivial logical, and weight-4 logicals exist (the stored witness), so d = 4 exactly for these generators; the SAT search reached the same exact value, and two further separate exhaustive programs agree. The board's Pauli-weight certifier is not available yet, so the claim is filed as an upper bound. The board's trusted gate (verify/validate_candidate.py) found no lighter logical in 3140 RIS trials (seed 1936550528), and the CI distance gate (verify/gate_changed.py, seed 1667834915) found none in 14 symmetry-orbit passes, 26920 deep RIS trials and 4038000 fast trials. Not CSS under any single-qubit Clifford: checked by the verifier on the generators and, by a separate linear-algebra test, on the whole stabilizer group. The board already has a CSS [[16,4,4]] of weight 6 (codes/16-4-4.json, @MathysRennela, found by locality-constrained SAT discovery); this is a different, non-CSS code, hence the -s slug. Literature check: we did not find a weight-5 [[16,4,4]] stabilizer code in Grassl's codetables.de (where the [[16,4]] optimum without a weight limit is d = 5, so d = 4 is not optimal there), in Wei, Han, He, Li and Liu (arXiv:2601.19848, tables n <= 12), in Wang, Liu, Li, Kubica and Gu (arXiv:2601.15446, whose [[16,4,4]] has weight 8) or in the EC Zoo (errorcorrectionzoo.org); literature novelty is not claimed.",
  "search_budget": {
   "cpu_hours": 4.4
  }
 },
 "family": "other"
}