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447, 654, 665, 675 ], [ 448, 449, 450, 451, 638, 666, 693 ], [ 452, 453, 454, 455, 667, 672, 676 ], [ 456, 457, 458, 459, 652, 668, 673, 677 ], [ 460, 461, 462, 463, 669, 671, 692 ], [ 464, 465, 466, 467, 675, 678, 680 ], [ 468, 469, 470, 471, 638, 639, 670, 681 ], [ 472, 473, 474, 475, 639, 668, 671, 690 ], [ 476, 477, 478, 479, 640, 669, 672, 691 ], [ 480, 481, 482, 483, 651, 667, 673, 678 ], [ 484, 485, 486, 487, 661, 670, 674, 692 ], [ 488, 489, 490, 491, 637, 662, 675 ], [ 492, 493, 494, 495, 659, 676, 691, 693 ], [ 496, 497, 498, 499, 651, 653, 677 ], [ 500, 501, 502, 503, 655, 678, 687 ], [ 504, 505, 506, 507, 643, 661, 679, 686 ], [ 508, 509, 510, 511, 644, 662, 680, 687 ], [ 512, 513, 514, 515, 645, 660, 681, 685 ], [ 516, 517, 518, 519, 664, 668, 682, 688 ], [ 520, 521, 522, 523, 636, 646, 669, 683 ], [ 524, 525, 526, 527, 665, 687, 689 ], [ 528, 529, 530, 531, 645, 648, 666, 684 ], [ 532, 533, 534, 535, 646, 685, 691 ], [ 536, 537, 538, 539, 650, 684, 686, 695 ], [ 540, 541, 542, 543, 642, 687 ], [ 544, 545, 546, 547, 643, 677, 688 ], [ 548, 549, 550, 551, 667, 675, 689 ], [ 552, 553, 554, 555, 677, 690, 694 ], [ 556, 557, 558, 559, 656, 691 ], [ 560, 561, 562, 563, 639, 658, 684, 692 ], [ 564, 565, 566, 567, 656, 660, 683, 693 ], [ 568, 569, 570, 571, 653, 664, 679, 694 ], [ 572, 573, 574, 575, 648, 670, 671, 695 ] ] }, "distance": { "d": 24, "X": { "value": 24, "confidence": "upper_bound", "witness": [ 52, 53, 60, 61, 84, 85, 92, 93, 120, 121, 148, 149, 312, 313, 372, 373, 392, 393, 488, 489, 512, 513, 528, 529 ] }, "Z": { "value": 24, "confidence": "upper_bound", "witness": [ 12, 13, 32, 33, 44, 45, 68, 69, 200, 201, 240, 241, 360, 361, 396, 397, 440, 441, 484, 485, 504, 505, 536, 537 ] } }, "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/144-24-12.json ([[144,24,12]] two-block code over C6 x C6 x C2; d_X = d_Z = 12 certified in certs/144-24-12.json), keeping all of its checks.X except the rows [3, 15, 17, 19, 24, 25, 30, 36, 37, 38, 44, 59] and all of its checks.Z except the rows [1, 6, 16, 17, 23, 34, 36, 45, 48, 55, 57, 66] (linearly independent, 60 and 60 rows). Qubits: central (q,a) -> 4q+a (q a base qubit, 0 <= a < 4); t_j -> 576+j for the j-th kept Z-check z_j; s_i -> 636+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..143); 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*144 + 60 + 60 = 696 = 5n - k of the base, k = 2*24 = 48, check weight 10.", "references": [ "arXiv:2609.37231", "arXiv:2610.13880", "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/144-24-12.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/144-24-12.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 = 48. 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" }