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[[832,68,10]] d ≤
n
832
k
68
d
10
kd²/n
8.173
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[5, 10, 23, 48, 97, 102, 115, 120, 157, 177]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[134, 139, 157, 184, 322, 327, 345, 380, 548, 556]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.308) · H_Z 2–3 (mean 2.308)
trapping sets H_X (1,2)×576 (2,2)×1152 (3,2)×2304 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 576 (1,3): 256 (2,2): 1152 (2,3): 2304 (2,4): 384 (3,2): 2304 (3,3): 11520 (3,4): 7296 (3,5): 3072 (3,6): 1152
trapping sets H_Z (1,2)×576 (2,2)×1152 (3,2)×2304 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 576 (1,3): 256 (2,2): 1152 (2,3): 2304 (2,4): 384 (3,2): 2304 (3,3): 11520 (3,4): 7296 (3,5): 3072 (3,6): 1152

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction Lifted product over F_2[G], G = Z_32 x|_17 Z_2 (order 64, metacyclic ZSZ(32,2,17)), with 2x3 base matrices of single group elements, A = [[18, 22, 38], [9, 40, 6]] (left regular representation) and B = [[22, 21, 5], [25, 61, 24]] (right regular representation) as element indices x^a y^b -> 2a + b; n = 13|G|, k = |G| + 4, check weight 5
model Claude Claude Opus 5.5 (claimed, not verified)
date 2026-10-18
family lifted product (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[832,68,10]] — weight-5 lifted product over F_2[Z_32 x|_17 Z_2] with 2x3 single-element base matrices

Direction & hypothesis

Cell: unrestricted x weight-6 (maximum check weight 5). The board's [[624,52,9]] is a lifted product with 2x3 base matrices whose entries are single group elements, so n = 13|G|, k = |G| + 4 and every check has weight 5 (three left-acting entries plus two right-acting ones). The hypothesis was that the same shape over slightly larger non-abelian groups keeps the distance at or above 9 while k grows with |G|; when this code was gated, the smallest distance that advanced the board at this (n, k, w) was 9.

What was searched

  • Lifted products from lifted_product_base in research/kit/nonabelian_lp.py

    • with 2x3 base matrices A (left regular representation) and B (right regular representation), every entry a single group element, over groups of order 47..76: ZSZ(l1, l2, q) metacyclic groups from zsz in the same module and the kit's small non-abelian groups. 6,744 connected candidates screened at 300 gf2_fast RIS trials (pair depth 8).

  • d = 10 appeared only at n = 780, 832 and 936. All seven candidates in these

    • three cells held d = 10 at 300, 5,000 and 300,000 trials; the best one per cell went to the gate and to fresh-seed ladders.

  • Dominance against the board checkout at run time (n down, k up, d up,

    • w down; equality counts as dominated).

Evidence trail

The submitted code: G = Z_32 x|_17 Z_2 of order 64 (x^32 = y^2 = 1, y x = x^17 y; element x^a y^b at index 2a + b), A = [[[18], [22], [38]], [[9], [40], [6]]] and B = [[[22], [21], [5]], [[25], [61], [24]]] as element indices. n = 13 x 64 = 832, k = 68 = |G| + 4, every check has weight 5.

| instrument (gf2_fast RIS, pair depth 8) | trials per side | lightest logical | |---|---:|---:| | screen stage 1 | 300 | 10 | | screen stage 2 | 5,000 | 10 | | screen stage 3 | 300,000 | 10 | | gate refutation pass (verify/validate_candidate.py) | gate's own | nothing lighter | | deep ladder, fresh seed 1300001 | 20,000,000 | 10 | | deep ladder, fresh seed 1307920 | 20,000,000 | 10 | | deep ladder, fresh seed 1315839 | 20,000,000 | 10 |

The reading never moved from 10 between 300 and 20,000,000 trials, and no rung found anything lighter. Claim: witness-backed upper bound d <= 10, not exact.

Dead ends

  • Shape A = [g1, 1 + g2], B = [1 + g3, g4] (check weight 5, n = 5|G|) over

    • groups of order 121..200: d = 3 in all 2,848 candidates at 300 trials.

  • In this 2x3 shape, d stayed at or below 9 for every other order in 47..76;

    • d = 10 needs the three orders above.

  • Abelian groups never exceeded d = 4 in the companion weight-6 sweep of the

    • same family (2,643 candidates), so the search kept to non-abelian groups.

Tools

Claude Opus 5.5 (provenance.model) on a Brev n2d-highcpu-64 box; gf2_fast (verify/gf2_fast.cpp) through research/kit/surrogate.py; research/kit/nonabelian_lp.py for the groups and the lifted product; research/kit/prescreen.py for the direct-sum check; research/kit/submit.py for packaging; verify/validate_candidate.py as the gate. About 2 CPU-hours of screening and about 25 CPU-hours of ladders for this code.

Reproduction

import sys; sys.path[:0] = ["research/kit", "verify"]
import nonabelian_lp as nlp
mul = nlp.zsz(32, 2, 17)
HX, HZ = nlp.lifted_product_base(mul, [[[18], [22], [38]], [[9], [40], [6]]], [[[22], [21], [5]], [[25], [61], [24]]])
assert HX.shape[1] == 832

Element indices follow nlp.zsz_index(32, 2, a, b) = 2a + b. Literature novelty unverified; board-relative novelty from verify/validate_candidate.py.

Parity checks

X-checks 384 (max weight 5) · Z-checks 384 (max weight 5)
H_X (384 checks, sparse supports)
[46, 234, 410, 598, 665] [47, 235, 411, 631, 664] [48, 236, 412, 600, 667] [49, 237, 413, 633, 666] [50, 238, 414, 602, 669] [51, 239, 415, 635, 668] [52, 240, 416, 604, 671] [53, 241, 417, 637, 670] [54, 242, 418, 606, 673] [55, 243, 419, 639, 672] [56, 244, 420, 608, 675] [57, 245, 421, 577, 674] [58, 246, 422, 610, 677] [59, 247, 423, 579, 676] [60, 248, 424, 612, 679] [61, 249, 425, 581, 678] [62, 250, 426, 614, 681] [63, 251, 427, 583, 680] [0, 252, 428, 616, 683] [1, 253, 429, 585, 682] [2, 254, 430, 618, 685] [3, 255, 431, 587, 684] [4, 192, 432, 620, 687] [5, 193, 433, 589, 686] [6, 194, 434, 622, 689] [7, 195, 435, 591, 688] [8, 196, 436, 624, 691] [9, 197, 437, 593, 690] [10, 198, 438, 626, 693] [11, 199, 439, 595, 692] [12, 200, 440, 628, 695] [13, 201, 441, 597, 694] [14, 202, 442, 630, 697] [15, 203, 443, 599, 696] [16, 204, 444, 632, 699] [17, 205, 445, 601, 698] [18, 206, 446, 634, 701] [19, 207, 447, 603, 700] [20, 208, 384, 636, 703] [21, 209, 385, 605, 702] [22, 210, 386, 638, 641] [23, 211, 387, 607, 640] [24, 212, 388, 576, 643] [25, 213, 389, 609, 642] [26, 214, 390, 578, 645] [27, 215, 391, 611, 644] [28, 216, 392, 580, 647] [29, 217, 393, 613, 646] [30, 218, 394, 582, 649] [31, 219, 395, 615, 648] [32, 220, 396, 584, 651] [33, 221, 397, 617, 650] [34, 222, 398, 586, 653] [35, 223, 399, 619, 652] [36, 224, 400, 588, 655] [37, 225, 401, 621, 654] [38, 226, 402, 590, 657] [39, 227, 403, 623, 656] [40, 228, 404, 592, 659] [41, 229, 405, 625, 658] [42, 230, 406, 594, 661] [43, 231, 407, 627, 660] [44, 232, 408, 596, 663] [45, 233, 409, 629, 662] [110, 298, 474, 597, 701] [111, 299, 475, 596, 700] [112, 300, 476, 599, 703] [113, 301, 477, 598, 702] [114, 302, 478, 601, 641] [115, 303, 479, 600, 640] [116, 304, 480, 603, 643] [117, 305, 481, 602, 642] [118, 306, 482, 605, 645] [119, 307, 483, 604, 644] [120, 308, 484, 607, 647] [121, 309, 485, 606, 646] [122, 310, 486, 609, 649] [123, 311, 487, 608, 648] [124, 312, 488, 611, 651] [125, 313, 489, 610, 650] [126, 314, 490, 613, 653] [127, 315, 491, 612, 652] [64, 316, 492, 615, 655] [65, 317, 493, 614, 654] [66, 318, 494, 617, 657] [67, 319, 495, 616, 656] [68, 256, 496, 619, 659] [69, 257, 497, 618, 658] [70, 258, 498, 621, 661] [71, 259, 499, 620, 660] [72, 260, 500, 623, 663] [73, 261, 501, 622, 662] [74, 262, 502, 625, 665] [75, 263, 503, 624, 664] [76, 264, 504, 627, 667] [77, 265, 505, 626, 666] [78, 266, 506, 629, 669] [79, 267, 507, 628, 668] [80, 268, 508, 631, 671] [81, 269, 509, 630, 670] [82, 270, 510, 633, 673] [83, 271, 511, 632, 672] [84, 272, 448, 635, 675] [85, 273, 449, 634, 674] [86, 274, 450, 637, 677] [87, 275, 451, 636, 676] [88, 276, 452, 639, 679] [89, 277, 453, 638, 678] [90, 278, 454, 577, 681] [91, 279, 455, 576, 680] [92, 280, 456, 579, 683] [93, 281, 457, 578, 682] [94, 282, 458, 581, 685] [95, 283, 459, 580, 684] [96, 284, 460, 583, 687] [97, 285, 461, 582, 686] [98, 286, 462, 585, 689] [99, 287, 463, 584, 688] [100, 288, 464, 587, 691] [101, 289, 465, 586, 690] [102, 290, 466, 589, 693] [103, 291, 467, 588, 692] [104, 292, 468, 591, 695] [105, 293, 469, 590, 694] [106, 294, 470, 593, 697] [107, 295, 471, 592, 696] [108, 296, 472, 595, 699] [109, 297, 473, 594, 698] [174, 362, 538, 581, 664] [175, 363, 539, 580, 665] [176, 364, 540, 583, 666] [177, 365, 541, 582, 667] [178, 366, 542, 585, 668] [179, 367, 543, 584, 669] [180, 368, 544, 587, 670] [181, 369, 545, 586, 671] [182, 370, 546, 589, 672] [183, 371, 547, 588, 673] [184, 372, 548, 591, 674] [185, 373, 549, 590, 675] [186, 374, 550, 593, 676] [187, 375, 551, 592, 677] [188, 376, 552, 595, 678] [189, 377, 553, 594, 679] [190, 378, 554, 597, 680] [191, 379, 555, 596, 681] [128, 380, 556, 599, 682] [129, 381, 557, 598, 683] [130, 382, 558, 601, 684] [131, 383, 559, 600, 685] [132, 320, 560, 603, 686] [133, 321, 561, 602, 687] [134, 322, 562, 605, 688] [135, 323, 563, 604, 689] [136, 324, 564, 607, 690] [137, 325, 565, 606, 691] [138, 326, 566, 609, 692] [139, 327, 567, 608, 693] [140, 328, 568, 611, 694] [141, 329, 569, 610, 695] [142, 330, 570, 613, 696] [143, 331, 571, 612, 697] [144, 332, 572, 615, 698] [145, 333, 573, 614, 699] [146, 334, 574, 617, 700] [147, 335, 575, 616, 701] [148, 336, 512, 619, 702] [149, 337, 513, 618, 703] [150, 338, 514, 621, 640] [151, 339, 515, 620, 641] [152, 340, 516, 623, 642] [153, 341, 517, 622, 643] [154, 342, 518, 625, 644] [155, 343, 519, 624, 645] [156, 344, 520, 627, 646] [157, 345, 521, 626, 647] [158, 346, 522, 629, 648] [159, 347, 523, 628, 649] [160, 348, 524, 631, 650] [161, 349, 525, 630, 651] [162, 350, 526, 633, 652] [163, 351, 527, 632, 653] [164, 352, 528, 635, 654] [165, 353, 529, 634, 655] [166, 354, 530, 637, 656] [167, 355, 531, 636, 657] [168, 356, 532, 639, 658] [169, 357, 533, 638, 659] [170, 358, 534, 577, 660] [171, 359, 535, 576, 661] [172, 360, 536, 579, 662] [173, 361, 537, 578, 663] [57, 216, 442, 726, 793] [56, 217, 443, 759, 792] [27, 218, 444, 728, 795] [26, 219, 445, 761, 794] [61, 220, 446, 730, 797] [60, 221, 447, 763, 796] [31, 222, 384, 732, 799] [30, 223, 385, 765, 798] [1, 224, 386, 734, 801] [0, 225, 387, 767, 800] [35, 226, 388, 736, 803] [34, 227, 389, 705, 802] [5, 228, 390, 738, 805] [4, 229, 391, 707, 804] [39, 230, 392, 740, 807] [38, 231, 393, 709, 806] [9, 232, 394, 742, 809] [8, 233, 395, 711, 808] [43, 234, 396, 744, 811] [42, 235, 397, 713, 810] [13, 236, 398, 746, 813] [12, 237, 399, 715, 812] [47, 238, 400, 748, 815] [46, 239, 401, 717, 814] [17, 240, 402, 750, 817] [16, 241, 403, 719, 816] [51, 242, 404, 752, 819] [50, 243, 405, 721, 818] [21, 244, 406, 754, 821] [20, 245, 407, 723, 820] [55, 246, 408, 756, 823] [54, 247, 409, 725, 822] [25, 248, 410, 758, 825] [24, 249, 411, 727, 824] [59, 250, 412, 760, 827] [58, 251, 413, 729, 826] [29, 252, 414, 762, 829] [28, 253, 415, 731, 828] [63, 254, 416, 764, 831] [62, 255, 417, 733, 830] [33, 192, 418, 766, 769] [32, 193, 419, 735, 768] [3, 194, 420, 704, 771] [2, 195, 421, 737, 770] [37, 196, 422, 706, 773] [36, 197, 423, 739, 772] [7, 198, 424, 708, 775] [6, 199, 425, 741, 774] [41, 200, 426, 710, 777] [40, 201, 427, 743, 776] [11, 202, 428, 712, 779] [10, 203, 429, 745, 778] [45, 204, 430, 714, 781] [44, 205, 431, 747, 780] [15, 206, 432, 716, 783] [14, 207, 433, 749, 782] [49, 208, 434, 718, 785] [48, 209, 435, 751, 784] [19, 210, 436, 720, 787] [18, 211, 437, 753, 786] [53, 212, 438, 722, 789] [52, 213, 439, 755, 788] [23, 214, 440, 724, 791] [22, 215, 441, 757, 790] [121, 280, 506, 725, 829] [120, 281, 507, 724, 828] [91, 282, 508, 727, 831] [90, 283, 509, 726, 830] [125, 284, 510, 729, 769] [124, 285, 511, 728, 768] [95, 286, 448, 731, 771] [94, 287, 449, 730, 770] [65, 288, 450, 733, 773] [64, 289, 451, 732, 772] [99, 290, 452, 735, 775] [98, 291, 453, 734, 774] [69, 292, 454, 737, 777] [68, 293, 455, 736, 776] [103, 294, 456, 739, 779] [102, 295, 457, 738, 778] [73, 296, 458, 741, 781] [72, 297, 459, 740, 780] [107, 298, 460, 743, 783] [106, 299, 461, 742, 782] [77, 300, 462, 745, 785] [76, 301, 463, 744, 784] [111, 302, 464, 747, 787] [110, 303, 465, 746, 786] [81, 304, 466, 749, 789] [80, 305, 467, 748, 788] [115, 306, 468, 751, 791] [114, 307, 469, 750, 790] [85, 308, 470, 753, 793] [84, 309, 471, 752, 792] [119, 310, 472, 755, 795] [118, 311, 473, 754, 794] [89, 312, 474, 757, 797] [88, 313, 475, 756, 796] [123, 314, 476, 759, 799] [122, 315, 477, 758, 798] [93, 316, 478, 761, 801] [92, 317, 479, 760, 800] [127, 318, 480, 763, 803] [126, 319, 481, 762, 802] [97, 256, 482, 765, 805] [96, 257, 483, 764, 804] [67, 258, 484, 767, 807] [66, 259, 485, 766, 806] [101, 260, 486, 705, 809] [100, 261, 487, 704, 808] [71, 262, 488, 707, 811] [70, 263, 489, 706, 810] [105, 264, 490, 709, 813] [104, 265, 491, 708, 812] [75, 266, 492, 711, 815] [74, 267, 493, 710, 814] [109, 268, 494, 713, 817] [108, 269, 495, 712, 816] [79, 270, 496, 715, 819] [78, 271, 497, 714, 818] [113, 272, 498, 717, 821] [112, 273, 499, 716, 820] [83, 274, 500, 719, 823] [82, 275, 501, 718, 822] [117, 276, 502, 721, 825] [116, 277, 503, 720, 824] [87, 278, 504, 723, 827] [86, 279, 505, 722, 826] [185, 344, 570, 709, 792] [184, 345, 571, 708, 793] [155, 346, 572, 711, 794] [154, 347, 573, 710, 795] [189, 348, 574, 713, 796] [188, 349, 575, 712, 797] [159, 350, 512, 715, 798] [158, 351, 513, 714, 799] [129, 352, 514, 717, 800] [128, 353, 515, 716, 801] [163, 354, 516, 719, 802] [162, 355, 517, 718, 803] [133, 356, 518, 721, 804] [132, 357, 519, 720, 805] [167, 358, 520, 723, 806] [166, 359, 521, 722, 807] [137, 360, 522, 725, 808] [136, 361, 523, 724, 809] [171, 362, 524, 727, 810] [170, 363, 525, 726, 811] [141, 364, 526, 729, 812] [140, 365, 527, 728, 813] [175, 366, 528, 731, 814] [174, 367, 529, 730, 815] [145, 368, 530, 733, 816] [144, 369, 531, 732, 817] [179, 370, 532, 735, 818] [178, 371, 533, 734, 819] [149, 372, 534, 737, 820] [148, 373, 535, 736, 821] [183, 374, 536, 739, 822] [182, 375, 537, 738, 823] [153, 376, 538, 741, 824] [152, 377, 539, 740, 825] [187, 378, 540, 743, 826] [186, 379, 541, 742, 827] [157, 380, 542, 745, 828] [156, 381, 543, 744, 829] [191, 382, 544, 747, 830] [190, 383, 545, 746, 831] [161, 320, 546, 749, 768] [160, 321, 547, 748, 769] [131, 322, 548, 751, 770] [130, 323, 549, 750, 771] [165, 324, 550, 753, 772] [164, 325, 551, 752, 773] [135, 326, 552, 755, 774] [134, 327, 553, 754, 775] [169, 328, 554, 757, 776] [168, 329, 555, 756, 777] [139, 330, 556, 759, 778] [138, 331, 557, 758, 779] [173, 332, 558, 761, 780] [172, 333, 559, 760, 781] [143, 334, 560, 763, 782] [142, 335, 561, 762, 783] [177, 336, 562, 765, 784] [176, 337, 563, 764, 785] [147, 338, 564, 767, 786] [146, 339, 565, 766, 787] [181, 340, 566, 705, 788] [180, 341, 567, 704, 789] [151, 342, 568, 707, 790] [150, 343, 569, 706, 791]
H_Z (384 checks, sparse supports)
[42, 109, 189, 594, 713] [11, 108, 188, 595, 712] [44, 111, 191, 596, 747] [13, 110, 190, 597, 746] [46, 113, 129, 598, 717] [15, 112, 128, 599, 716] [48, 115, 131, 600, 751] [17, 114, 130, 601, 750] [50, 117, 133, 602, 721] [19, 116, 132, 603, 720] [52, 119, 135, 604, 755] [21, 118, 134, 605, 754] [54, 121, 137, 606, 725] [23, 120, 136, 607, 724] [56, 123, 139, 608, 759] [25, 122, 138, 609, 758] [58, 125, 141, 610, 729] [27, 124, 140, 611, 728] [60, 127, 143, 612, 763] [29, 126, 142, 613, 762] [62, 65, 145, 614, 733] [31, 64, 144, 615, 732] [0, 67, 147, 616, 767] [33, 66, 146, 617, 766] [2, 69, 149, 618, 737] [35, 68, 148, 619, 736] [4, 71, 151, 620, 707] [37, 70, 150, 621, 706] [6, 73, 153, 622, 741] [39, 72, 152, 623, 740] [8, 75, 155, 624, 711] [41, 74, 154, 625, 710] [10, 77, 157, 626, 745] [43, 76, 156, 627, 744] [12, 79, 159, 628, 715] [45, 78, 158, 629, 714] [14, 81, 161, 630, 749] [47, 80, 160, 631, 748] [16, 83, 163, 632, 719] [49, 82, 162, 633, 718] [18, 85, 165, 634, 753] [51, 84, 164, 635, 752] [20, 87, 167, 636, 723] [53, 86, 166, 637, 722] [22, 89, 169, 638, 757] [55, 88, 168, 639, 756] [24, 91, 171, 576, 727] [57, 90, 170, 577, 726] [26, 93, 173, 578, 761] [59, 92, 172, 579, 760] [28, 95, 175, 580, 731] [61, 94, 174, 581, 730] [30, 97, 177, 582, 765] [63, 96, 176, 583, 764] [32, 99, 179, 584, 735] [1, 98, 178, 585, 734] [34, 101, 181, 586, 705] [3, 100, 180, 587, 704] [36, 103, 183, 588, 739] [5, 102, 182, 589, 738] [38, 105, 185, 590, 709] [7, 104, 184, 591, 708] [40, 107, 187, 592, 743] [9, 106, 186, 593, 742] [41, 69, 168, 658, 777] [40, 68, 169, 659, 776] [43, 71, 170, 660, 811] [42, 70, 171, 661, 810] [45, 73, 172, 662, 781] [44, 72, 173, 663, 780] [47, 75, 174, 664, 815] [46, 74, 175, 665, 814] [49, 77, 176, 666, 785] [48, 76, 177, 667, 784] [51, 79, 178, 668, 819] [50, 78, 179, 669, 818] [53, 81, 180, 670, 789] [52, 80, 181, 671, 788] [55, 83, 182, 672, 823] [54, 82, 183, 673, 822] [57, 85, 184, 674, 793] [56, 84, 185, 675, 792] [59, 87, 186, 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Code ID 832-68-10 · download JSON · raw on GitHub