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729, 730, 731, 832, 850 ], [ 732, 733, 734, 735, 833, 851 ], [ 736, 737, 738, 739, 834, 852 ], [ 740, 741, 742, 743, 835, 853 ] ] }, "distance": { "d": 12, "X": { "value": 12, "confidence": "exact", "witness": [ 28, 29, 196, 197, 312, 313, 500, 501, 724, 725, 740, 741 ] }, "Z": { "value": 12, "confidence": "exact", "witness": [ 0, 1, 124, 125, 324, 325, 428, 429, 572, 573, 676, 677 ] } }, "provenance": { "authors": [ "@TimoKellerMath" ], "origin": "submission", "novelty": "unknown", "construction": "One [[4,2,2]] distance-amplification step (Liang, Gu, Chen, Eisert, Wang, arXiv:2609.37231, Eqs. (14), (15); amplifier checks G_X = G_Z = the all-one row of length 4) applied to the board entry codes/186-64-6.json ([[186,64,6]] code of Construction I of arXiv:2610.19451; d_X = d_Z = 6 certified in certs/186-64-6.json), keeping all of its checks.X except the rows [61] and all of its checks.Z except the rows [61] (linearly independent, 61 and 61 rows). Qubits: central (q,a) -> 4q+a (q a base qubit, 0 <= a < 4); t_j -> 744+j for the j-th kept Z-check z_j; s_i -> 805+i for the i-th kept X-check c_i. X-checks: (c_i,a) = {4q+a : q in c_i} + {s_i} (i, then a), then (q) = {4q,...,4q+3} + {t_j : q in z_j} (q = 0..185); Z-checks: (z_j,a) = {4q+a : q in z_j} + {t_j}, then (q) = {4q,...,4q+3} + {s_i : q in c_i}. n = 4*186 + 61 + 61 = 866 = 5n - k of the base, k = 2*64 = 128, check weight 7.", "references": [ "arXiv:2609.37231", "arXiv:2610.19451", "arXiv:1512.07081", "arXiv:2611.08247" ], "date": "2026-11-10", "model": "Claude Opus 5.5", "notes": "Distance: d_X = d_Z = 12 exactly: d_P = 2 d_P(base) for P = X, Z by Lemma 4 of arXiv:2609.37231 (proof in the coordinates of this entry: Proposition 4 of amplified-and-folded-codes/README.md, github.com/TimoKellerMath/quantum-error-correction @ c203901854da), and d_X = d_Z = 6 for the base is certified in certs/186-64-6.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/186-64-6.json. d = 12 is also certified directly by tropidist b03ec2d4077f in certs/866-128-12.json (same pull request). The construction and the base are published; at board commit 8042f9da no CSS entry dominates or equals this code in (n, k, d, w), and no entry has its n and k unless named here; checked, not equivalent to an existing entry." }, "family": "distance-amplified" }