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454, 455, 601, 604, 607, 611 ], [ 456, 457, 458, 459, 460, 461, 602, 605, 609, 613 ], [ 462, 463, 464, 465, 466, 467, 591, 603, 606, 608, 610, 612 ], [ 468, 469, 470, 471, 472, 473, 586, 601, 604, 607, 613, 615 ], [ 474, 475, 476, 477, 478, 479, 587, 602, 605, 612 ], [ 480, 481, 482, 483, 484, 485, 603, 606, 608, 614 ], [ 486, 487, 488, 489, 490, 491, 579, 585, 596, 609, 612, 614 ], [ 492, 493, 494, 495, 496, 497, 583, 597, 610, 615 ], [ 498, 499, 500, 501, 502, 503, 578, 584, 595, 611, 613 ], [ 504, 505, 506, 507, 508, 509, 581, 599, 609, 612, 614 ], [ 510, 511, 512, 513, 514, 515, 578, 582, 600, 610, 615 ], [ 516, 517, 518, 519, 520, 521, 579, 580, 598, 611, 613 ], [ 522, 523, 524, 525, 526, 527, 582, 584, 593, 609, 612, 614 ], [ 528, 529, 530, 531, 532, 533, 580, 585, 594, 610, 615 ], [ 534, 535, 536, 537, 538, 539, 581, 583, 592, 611, 613 ] ] }, "distance": { "d": 22, "X": { "value": 22, "confidence": "upper_bound", "witness": [ 78, 79, 168, 169, 210, 211, 246, 247, 264, 265, 342, 343, 348, 349, 420, 421, 462, 463, 492, 493, 498, 499 ] }, "Z": { "value": 22, "confidence": "upper_bound", "witness": [ 42, 43, 108, 109, 120, 121, 132, 133, 222, 223, 228, 229, 258, 259, 396, 397, 426, 427, 450, 451, 492, 493 ] } }, "provenance": { "authors": [ "@TimoKellerMath" ], "origin": "submission", "novelty": "unknown", "construction": "One [[6,4,2]] 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 6) applied to the board entry codes/90-14-11.json ([[90,14,11]] two-block code over a group of order 45; d_X = d_Z = 11 certified in certs/90-14-11.json), keeping all of its checks.X except the rows [2, 9, 16, 17, 34, 40, 44] and all of its checks.Z except the rows [3, 9, 21, 23, 30, 39, 44] (linearly independent, 38 and 38 rows). Qubits: central (q,a) -> 6q+a (q a base qubit, 0 <= a < 6); t_j -> 540+j for the j-th kept Z-check z_j; s_i -> 578+i for the i-th kept X-check c_i. X-checks: (c_i,a) = {6q+a : q in c_i} + {s_i} (i, then a), then (q) = {6q,...,6q+5} + {t_j : q in z_j} (q = 0..89); Z-checks: (z_j,a) = {6q+a : q in z_j} + {t_j}, then (q) = {6q,...,6q+5} + {s_i : q in c_i}. n = 6*90 + 38 + 38 = 616 = 7n - k of the base, k = 4*14 = 56, check weight 12.", "references": [ "arXiv:2609.37231", "arXiv:1512.07081", "arXiv:2611.08247" ], "date": "2026-11-10", "model": "Claude Opus 5.5", "notes": "Distance: d_X = d_Z = 22 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 = 11 for the base is certified in certs/90-14-11.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/90-14-11.json. Filed as upper_bound with these witnesses: the board earns the exact tier only through certs/, and its certifiers do not reach d = 22 at k = 56. 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" }