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 },
 "distance": {
  "d": 28,
  "P": {
   "value": 28,
   "confidence": "upper_bound",
   "witness": {
    "X": [
     23,
     57,
     66,
     86,
     100,
     146,
     160,
     234,
     248,
     254,
     268,
     314,
     328
    ],
    "Z": [
     37,
     38,
     51,
     52,
     78,
     86,
     100,
     143,
     157,
     206,
     219,
     220,
     232,
     246,
     254,
     268,
     271,
     311,
     325
    ]
   }
  }
 },
 "provenance": {
  "authors": [
   "@MicheldeMessieres"
  ],
  "construction": "Symplectic halving (mirror fold) of the board's CSS entry codes/672-20-32.json ([[672,20,32]], by @vprusso), a two-block group algebra (2BGA) code over the metacyclic group C_12 semidirect_5 C_28 of order 336 (construction of arXiv:2306.16400, Lin and Pryadko). The fold uses a fixed-point-free involution tau of the parent's qubits that maps the X-check rowspace onto the Z-check rowspace: tau(1,g) = (2, h*psi(g)) with h = x^2 y^3 and psi the automorphism x -> x^7, y -> y^-1 (elements x^i y^j indexed i*28+j as in the parent); block 1 = parent qubits 0..335, block 2 = 336..671. Fold qubit i is the pair {q_i, tau(q_i)}, q_i the i-th smallest qubit with q_i < tau(q_i). Each parent X-check row gives one generator: X on the pairs whose q_i is in the row, Z on the pairs whose tau(q_i) is in the row. S = (A|B) is isotropic, k = 20/2 = 10, check weight 6, not CSS under any local Clifford.",
  "origin": "submission",
  "date": "2026-10-11",
  "model": "Claude Opus 5.5 (Claude Code)",
  "notes": "Derived from codes/672-20-32.json by @vprusso (two-block group algebra (2BGA) code over the metacyclic group C_12 semidirect_5 C_28 of order 336 (construction of arXiv:2306.16400, Lin and Pryadko)). The halving/doubling correspondence is the board's fieldnote fieldnotes/2026-09-28-symplectic-doubling-and-halving.md (@MathysRennela; Prop. 4 of arXiv:2609.30069), and the abelian mirror-code construction of Khesin and Lu (arXiv:2603.05496); mirror folds of bicycle codes are already on the board from @mrvee-qC-bee (codes/155-6-21.json, codes/155-6-23.json, codes/144-10-16-b.json) and CPM halvings from @MathysRennela (codes/145-32-6.json, codes/188-49-6.json). Our part, stated narrowly: we enumerated the X<->Z mirrors that are automorphisms of the parent's Tanner graph (fixed-point-free involutions swapping the X and Z checks, up to conjugacy; nauty + GAP), folded along each, and kept the mirror whose lightest logical found stayed heaviest; then ran a deep distance search on the fold (2.8 CPU-h on this fold: orbit trick: 1.5 CPU-h, 934670 trials; frame-randomised Pauli RIS: 0.9 CPU-h, 523469 trials; symmetry-restricted SAT: 0.3 CPU-h; exact SAT floor ladder: 0.1 CPU-h; light logicals shared between mirrors of the same parent). Distance d = 28 is the Pauli weight of the lightest logical any method found: a witness-backed upper bound, not an exact certificate. Choosing the strongest of several mirrors by their upper bounds can favour a mirror that was simply searched less well; see the note. Construction and literature novelty are not claimed.",
  "references": [
   "arXiv:2306.16400",
   "arXiv:2609.30069",
   "arXiv:2603.05496"
  ],
  "novelty": "unknown",
  "search_budget": {
   "cpu_hours": 2.8,
   "tool": "nauty+GAP mirror enumeration; orbit trick, frame-randomised Pauli RIS, symmetric SAT (local scripts)",
   "notes": "CPU-hours spent on this fold's distance search after the mirror screen; excludes the 40-parent screen (about 3.5 CPU-h) and the board gate runs."
  }
 },
 "family": "2bga-coset"
}