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648, 649, 650, 651, 816, 863 ], [ 652, 653, 654, 655, 823, 849, 864 ], [ 656, 657, 658, 659, 818, 852, 865 ], [ 660, 661, 662, 663, 800, 856, 866 ], [ 664, 665, 666, 667, 821, 827, 867 ], [ 668, 669, 670, 671, 837, 868, 879 ], [ 672, 673, 674, 675, 810, 855, 869 ], [ 676, 677, 678, 679, 828, 840, 870 ], [ 680, 681, 682, 683, 812, 871 ], [ 684, 685, 686, 687, 801, 850, 872 ], [ 688, 689, 690, 691, 834, 859, 873 ], [ 692, 693, 694, 695, 861 ], [ 696, 697, 698, 699, 802, 830, 874 ], [ 700, 701, 702, 703, 843, 875 ], [ 704, 705, 706, 707, 822, 867, 876 ], [ 708, 709, 710, 711, 805, 825, 877 ], [ 712, 713, 714, 715, 808, 845, 878 ], [ 716, 717, 718, 719, 814, 836, 879 ] ] }, "distance": { "d": 22, "X": { "value": 22, "confidence": "upper_bound", "witness": [ 28, 29, 140, 141, 144, 145, 328, 329, 444, 445, 524, 525, 528, 529, 588, 589, 596, 597, 640, 641, 648, 649 ] }, "Z": { "value": 22, "confidence": "upper_bound", "witness": [ 64, 65, 204, 205, 256, 257, 260, 261, 268, 269, 304, 305, 312, 313, 412, 413, 552, 553, 560, 561, 600, 601 ] } }, "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/180-20-11.json ([[180,20,11]] two-block code over C15 x S3; d_X = d_Z = 11 certified in certs/180-20-11.json), keeping all of its checks.X except the rows [20, 28, 30, 31, 54, 59, 60, 65, 68, 83] and all of its checks.Z except the rows [17, 18, 22, 40, 46, 52, 74, 79, 82, 85] (linearly independent, 80 and 80 rows). Qubits: central (q,a) -> 4q+a (q a base qubit, 0 <= a < 4); t_j -> 720+j for the j-th kept Z-check z_j; s_i -> 800+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..179); 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*180 + 80 + 80 = 880 = 5n - k of the base, k = 2*20 = 40, check weight 8.", "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 = 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/180-20-11.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/180-20-11.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 = 40. 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" }