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 "distance": {
  "d": 12,
  "X": {
   "value": 12,
   "confidence": "upper_bound",
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 "provenance": {
  "authors": [
   "Yang, Willers",
   "Duckering, Casey",
   "Dua, Arpit"
  ],
  "construction": "GALA code (group-action lift with active orthogonality), L=12, J=5, group C_11 (trivial non-abelian factor). Block-circulant parents Hhat_X=[F|G], Hhat_Z=[G^T|F^T] over F_2[Z_6 x C_11] with F=(x^2,x^4,x^3,x^6,x^3,x^9), G=(x^9,x^2,x^8,x^5,x^8,x^7); the first J=5 of L/2=6 block rows are kept as H_X, H_Z. Self-dual (H_X=H_Z), girth 4 (660 4-cycles), weight 12.",
  "origin": "baseline",
  "date": "2026-08-07",
  "references": [
   "arXiv:2608.07431"
  ],
  "notes": "Distance d=12 certified exactly in arXiv:2608.07431 (Table S5; exhaustive exclusion of all lower-weight logicals plus an explicit weight-12 witness). Rebuilt here from the published generators \u2014 no parity-check matrices are distributed with the preprint; the rebuild reproduces the paper's rank(H)=51, k=30, weight 12, self-duality (H_X=H_Z) and 4-cycle count t_4=660. Witnesses recorded here are decoder-found weight-12 logicals (upper bound)."
 },
 "family": "generalized-bicycle"
}