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628, 648, 649 ], [ 378, 379, 380, 381, 382, 383, 616, 618, 629, 652 ], [ 384, 385, 386, 387, 388, 389, 619, 630, 651, 652 ], [ 390, 391, 392, 393, 394, 395, 617, 620, 631, 651 ], [ 396, 397, 398, 399, 400, 401, 618, 621, 654, 655 ], [ 402, 403, 404, 405, 406, 407, 619, 622, 632, 653, 655 ], [ 408, 409, 410, 411, 412, 413, 620, 623, 633, 653, 654 ], [ 414, 415, 416, 417, 418, 419, 621, 634, 647 ], [ 420, 421, 422, 423, 424, 425, 622, 624, 635, 646 ], [ 426, 427, 428, 429, 430, 431, 623, 625, 636, 646, 647 ], [ 432, 433, 434, 435, 436, 437, 619, 620, 626, 634, 637 ], [ 438, 439, 440, 441, 442, 443, 618, 620, 627, 635 ], [ 444, 445, 446, 447, 448, 449, 618, 619, 628, 636, 638 ], [ 450, 451, 452, 453, 454, 455, 622, 623, 626, 629, 639 ], [ 456, 457, 458, 459, 460, 461, 621, 623, 627, 630, 640 ], [ 462, 463, 464, 465, 466, 467, 621, 622, 628, 631, 641 ], [ 468, 469, 470, 471, 472, 473, 624, 625, 629, 642 ], [ 474, 475, 476, 477, 478, 479, 625, 630, 632, 643 ], [ 480, 481, 482, 483, 484, 485, 624, 631, 633 ], [ 486, 487, 488, 489, 490, 491, 617, 634 ], [ 492, 493, 494, 495, 496, 497, 616, 617, 632, 635, 644 ], [ 498, 499, 500, 501, 502, 503, 616, 633, 636, 645 ], [ 504, 505, 506, 507, 508, 509, 630, 631, 637, 646 ], [ 510, 511, 512, 513, 514, 515, 629, 631, 644, 647 ], [ 516, 517, 518, 519, 520, 521, 629, 630, 638, 645 ], [ 522, 523, 524, 525, 526, 527, 632, 633, 637, 639, 648 ], [ 528, 529, 530, 531, 532, 533, 633, 640, 649 ], [ 534, 535, 536, 537, 538, 539, 632, 638, 641, 650 ], [ 540, 541, 542, 543, 544, 545, 635, 636, 639, 642, 651 ], [ 546, 547, 548, 549, 550, 551, 634, 636, 640, 643 ], [ 552, 553, 554, 555, 556, 557, 634, 635, 641, 652 ], [ 558, 559, 560, 561, 562, 563, 627, 628, 642, 653 ], [ 564, 565, 566, 567, 568, 569, 626, 628, 643, 644, 654 ], [ 570, 571, 572, 573, 574, 575, 626, 627, 645, 655 ] ] }, "distance": { "d": 24, "X": { "value": 24, "confidence": "upper_bound", "witness": [ 6, 7, 24, 25, 42, 43, 198, 199, 228, 229, 270, 271, 282, 283, 336, 337, 366, 367, 504, 505, 528, 529, 546, 547 ] }, "Z": { "value": 24, "confidence": "upper_bound", "witness": [ 144, 145, 156, 157, 174, 175, 192, 193, 258, 259, 324, 325, 330, 331, 342, 343, 462, 463, 474, 475, 504, 505, 540, 541 ] } }, "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/96-16-12.json ([[96,16,12]] two-block code over a group of order 48; d_X = d_Z = 12 certified in certs/96-16-12.json), keeping all of its checks.X except the rows [1, 9, 18, 25, 32, 33, 38, 43] and all of its checks.Z except the rows [4, 5, 6, 11, 40, 41, 42, 47] (linearly independent, 40 and 40 rows). Qubits: central (q,a) -> 6q+a (q a base qubit, 0 <= a < 6); t_j -> 576+j for the j-th kept Z-check z_j; s_i -> 616+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..95); 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*96 + 40 + 40 = 656 = 7n - k of the base, k = 4*16 = 64, check weight 11.", "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 = 24 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 = 12 for the base is certified in certs/96-16-12.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/96-16-12.json. Filed as upper_bound with these witnesses: the board earns the exact tier only through certs/, and its certifiers do not reach d = 24 at k = 64. 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" }