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

Share this result

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)
[0, 43, 47, 60, 64, 103, 107, 124, 167, 175]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[0, 3, 9, 15, 192, 195, 207, 209, 393, 401]
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 1–3 (mean 2.296) · H_Z 1–3 (mean 2.296)
trapping sets H_X (1,1)×7 (2,1)×15 (3,1)×33 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 7 (1,2): 572 (1,3): 253 (2,1): 15 (2,2): 1159 (2,3): 2269 (2,4): 377 (3,1): 33 (3,2): 2400 (3,3): 11363 (3,4): 7175 (3,5): 3020 (3,6): 1125
trapping sets H_Z (1,1)×7 (2,1)×15 (3,1)×33 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 7 (1,2): 572 (1,3): 253 (2,1): 15 (2,2): 1159 (2,3): 2269 (2,4): 377 (3,1): 33 (3,2): 2400 (3,3): 11363 (3,4): 7175 (3,5): 3020 (3,6): 1125

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction Group from literal permutation generators [[[1, 9], [2, 20], [3, 11], [4, 22], [5, 13], [6, 24], [7, 15], [8, 26], [10, 28], [12, 30], [14, 32], [16, 18], [17, 25], [19, 27], [21, 29], [23, 31]], [[1, 27], [2, 28], [3, 29], [4, 30], [5, 31], [6, 32], [7, 17], [8, 18], [9, 19], [10, 20], [11, 21], [12, 22], [13, 23], [14, 24], [15, 25], [16, 26]], [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]]]. Generate closure from identity by right multiplication in the listed generator order, composition a*b=a after b; index by first discovery. Source https://people.maths.bris.ac.uk/~matyd/GroupNames/61/C2%5E2sC16.html. Left seed A=[[I,I,I],[I,L_3,L_13]], right seed B=[[I,I,I],[I,R_3,R_25]]. L_g[t,u]=1 iff u=g*t; R_h[t,u]=1 iff u=t*h. Product X(i,j) has A(i,k) on VV(k,j), B(l,j) on CC(i,l). Z(k,l) has transpose(B(l,j)) on VV(k,j), transpose(A(i,k)) on CC(i,l). Keep first independent rows in each sector.
model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)
builds on https://people.maths.bris.ac.uk/~matyd/GroupNames/61/C2%5E2sC16.html, https://arxiv.org/abs/2111.03654, https://arxiv.org/abs/2607.28795, https://arxiv.org/abs/1611.07164
date 2026-10-17
notes Displayed presentation: maximum check weight 5, maximum qubit degree 6, 764 checks and total incidence 3820.
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]] - CSS sparse stabilizer code

Direction & hypothesis

Study sparse finite instances of the construction below and their dimension–distance tradeoff.

What was searched

Group from literal permutation generators [[[1, 9], [2, 20], [3, 11], [4, 22], [5, 13], [6, 24], [7, 15], [8, 26], [10, 28], [12, 30], [14, 32], [16, 18], [17, 25], [19, 27], [21, 29], [23, 31]], [[1, 27], [2, 28], [3, 29], [4, 30], [5, 31], [6, 32], [7, 17], [8, 18], [9, 19], [10, 20], [11, 21], [12, 22], [13, 23], [14, 24], [15, 25], [16, 26]], [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]]]. Generate closure from identity by right multiplication in the listed generator order, composition a*b=a after b; index by first discovery. Source https://people.maths.bris.ac.uk/~matyd/GroupNames/61/C2%5E2sC16.html. Left seed A=[[I,I,I],[I,L_3,L_13]], right seed B=[[I,I,I],[I,R_3,R_25]]. L_g[t,u]=1 iff u=g*t; R_h[t,u]=1 iff u=t*h. Product X(i,j) has A(i,k) on VV(k,j), B(l,j) on CC(i,l). Z(k,l) has transpose(B(l,j)) on VV(k,j), transpose(A(i,k)) on CC(i,l). Keep first independent rows in each sector.

The displayed presentation has maximum check weight 5, maximum qubit degree 6, 764 checks and total incidence 3820.

Evidence trail

The JSON contains weight-10 logical witnesses. Distance confidence: X: upper_bound, Z: upper_bound.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. Official tooling: verify/qldpc_verify.py.

Reproduction

Read codes/832-68-10-group64.json. Allocate zero binary matrices H_X and H_Z with 832 columns. For each row list in checks.X or checks.Z, set the listed zero-based columns to 1, preserving row order. This reconstructs the displayed presentation exactly.

Construction references:

  • https://people.maths.bris.ac.uk/~matyd/GroupNames/61/C2%5E2sC16.html
  • https://arxiv.org/abs/2111.03654
  • https://arxiv.org/abs/2607.28795
  • https://arxiv.org/abs/1611.07164

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Parity checks

X-checks 382 (max weight 5) · Z-checks 382 (max weight 5)
H_X (382 checks, sparse supports)
[0, 192, 384, 576, 640] [1, 193, 385, 577, 641] [2, 194, 386, 578, 642] [3, 195, 387, 579, 643] [4, 196, 388, 580, 644] [5, 197, 389, 581, 645] [6, 198, 390, 582, 646] [7, 199, 391, 583, 647] [8, 200, 392, 584, 648] [9, 201, 393, 585, 649] [10, 202, 394, 586, 650] [11, 203, 395, 587, 651] [12, 204, 396, 588, 652] [13, 205, 397, 589, 653] [14, 206, 398, 590, 654] [15, 207, 399, 591, 655] [16, 208, 400, 592, 656] [17, 209, 401, 593, 657] [18, 210, 402, 594, 658] [19, 211, 403, 595, 659] [20, 212, 404, 596, 660] [21, 213, 405, 597, 661] [22, 214, 406, 598, 662] [23, 215, 407, 599, 663] [24, 216, 408, 600, 664] [25, 217, 409, 601, 665] [26, 218, 410, 602, 666] [27, 219, 411, 603, 667] [28, 220, 412, 604, 668] [29, 221, 413, 605, 669] [30, 222, 414, 606, 670] [31, 223, 415, 607, 671] [32, 224, 416, 608, 672] [33, 225, 417, 609, 673] [34, 226, 418, 610, 674] [35, 227, 419, 611, 675] [36, 228, 420, 612, 676] [37, 229, 421, 613, 677] [38, 230, 422, 614, 678] [39, 231, 423, 615, 679] [40, 232, 424, 616, 680] [41, 233, 425, 617, 681] [42, 234, 426, 618, 682] [43, 235, 427, 619, 683] [44, 236, 428, 620, 684] [45, 237, 429, 621, 685] [46, 238, 430, 622, 686] [47, 239, 431, 623, 687] [48, 240, 432, 624, 688] [49, 241, 433, 625, 689] [50, 242, 434, 626, 690] [51, 243, 435, 627, 691] [52, 244, 436, 628, 692] [53, 245, 437, 629, 693] [54, 246, 438, 630, 694] [55, 247, 439, 631, 695] [56, 248, 440, 632, 696] [57, 249, 441, 633, 697] [58, 250, 442, 634, 698] [59, 251, 443, 635, 699] [60, 252, 444, 636, 700] [61, 253, 445, 637, 701] [62, 254, 446, 638, 702] [63, 255, 447, 639, 703] [64, 256, 448, 576, 643] [65, 257, 449, 577, 645] [66, 258, 450, 578, 646] [67, 259, 451, 579, 648] [68, 260, 452, 580, 647] [69, 261, 453, 581, 649] [70, 262, 454, 582, 650] [71, 263, 455, 583, 651] [72, 264, 456, 584, 652] [73, 265, 457, 585, 653] [74, 266, 458, 586, 654] [75, 267, 459, 587, 655] [76, 268, 460, 588, 656] [77, 269, 461, 589, 657] [78, 270, 462, 590, 658] [79, 271, 463, 591, 659] [80, 272, 464, 592, 660] [81, 273, 465, 593, 661] [82, 274, 466, 594, 662] [83, 275, 467, 595, 663] [84, 276, 468, 596, 664] [85, 277, 469, 597, 665] [86, 278, 470, 598, 666] [87, 279, 471, 599, 667] [88, 280, 472, 600, 668] [89, 281, 473, 601, 669] [90, 282, 474, 602, 670] [91, 283, 475, 603, 671] [92, 284, 476, 604, 672] [93, 285, 477, 605, 673] [94, 286, 478, 606, 674] [95, 287, 479, 607, 675] [96, 288, 480, 608, 676] [97, 289, 481, 609, 677] [98, 290, 482, 610, 678] [99, 291, 483, 611, 679] [100, 292, 484, 612, 680] [101, 293, 485, 613, 681] [102, 294, 486, 614, 682] [103, 295, 487, 615, 683] [104, 296, 488, 616, 684] [105, 297, 489, 617, 685] [106, 298, 490, 618, 686] [107, 299, 491, 619, 687] [108, 300, 492, 620, 688] [109, 301, 493, 621, 689] [110, 302, 494, 622, 690] [111, 303, 495, 623, 691] [112, 304, 496, 624, 692] [113, 305, 497, 625, 693] [114, 306, 498, 626, 694] [115, 307, 499, 627, 695] [116, 308, 500, 628, 696] [117, 309, 501, 629, 697] [118, 310, 502, 630, 698] [119, 311, 503, 631, 699] [120, 312, 504, 632, 700] [121, 313, 505, 633, 701] [122, 314, 506, 634, 702] [123, 315, 507, 635, 703] [124, 316, 508, 636, 640] [125, 317, 509, 637, 641] [126, 318, 510, 638, 642] [127, 319, 511, 639, 644] [128, 320, 512, 576, 665] [129, 321, 513, 577, 664] [130, 322, 514, 578, 667] [131, 323, 515, 579, 671] [132, 324, 516, 580, 666] [133, 325, 517, 581, 670] [134, 326, 518, 582, 669] [135, 327, 519, 583, 668] [136, 328, 520, 584, 673] [137, 329, 521, 585, 672] [138, 330, 522, 586, 675] [139, 331, 523, 587, 674] [140, 332, 524, 588, 679] [141, 333, 525, 589, 678] [142, 334, 526, 590, 677] [143, 335, 527, 591, 676] [144, 336, 528, 592, 681] [145, 337, 529, 593, 680] [146, 338, 530, 594, 683] [147, 339, 531, 595, 682] [148, 340, 532, 596, 687] [149, 341, 533, 597, 686] [150, 342, 534, 598, 685] [151, 343, 535, 599, 684] [152, 344, 536, 600, 689] [153, 345, 537, 601, 688] [154, 346, 538, 602, 691] [155, 347, 539, 603, 690] [156, 348, 540, 604, 695] [157, 349, 541, 605, 694] [158, 350, 542, 606, 693] [159, 351, 543, 607, 692] [160, 352, 544, 608, 697] [161, 353, 545, 609, 696] [162, 354, 546, 610, 699] [163, 355, 547, 611, 698] [164, 356, 548, 612, 703] [165, 357, 549, 613, 702] [166, 358, 550, 614, 701] [167, 359, 551, 615, 700] [168, 360, 552, 616, 641] [169, 361, 553, 617, 640] [170, 362, 554, 618, 644] [171, 363, 555, 619, 642] [172, 364, 556, 620, 647] [173, 365, 557, 621, 646] [174, 366, 558, 622, 645] [175, 367, 559, 623, 643] [176, 368, 560, 624, 649] [177, 369, 561, 625, 648] [178, 370, 562, 626, 651] [179, 371, 563, 627, 650] [180, 372, 564, 628, 655] [181, 373, 565, 629, 654] [182, 374, 566, 630, 653] [183, 375, 567, 631, 652] [184, 376, 568, 632, 657] [185, 377, 569, 633, 656] [186, 378, 570, 634, 659] [187, 379, 571, 635, 658] [188, 380, 572, 636, 663] [189, 381, 573, 637, 662] [190, 382, 574, 638, 661] [191, 383, 575, 639, 660] [0, 195, 397, 704, 768] [1, 199, 398, 705, 769] [2, 198, 399, 706, 770] [3, 200, 401, 707, 771] [4, 197, 396, 708, 772] [5, 203, 402, 709, 773] [6, 202, 403, 710, 774] [7, 201, 400, 711, 775] [8, 204, 405, 712, 776] [9, 207, 406, 713, 777] [10, 206, 407, 714, 778] [11, 205, 404, 715, 779] [12, 208, 409, 716, 780] [13, 211, 410, 717, 781] [14, 210, 411, 718, 782] [15, 209, 408, 719, 783] [16, 212, 413, 720, 784] [17, 215, 414, 721, 785] [18, 214, 415, 722, 786] [19, 213, 412, 723, 787] [20, 216, 417, 724, 788] [21, 219, 418, 725, 789] [22, 218, 419, 726, 790] [23, 217, 416, 727, 791] [24, 220, 421, 728, 792] [25, 223, 422, 729, 793] [26, 222, 423, 730, 794] [27, 221, 420, 731, 795] [28, 224, 425, 732, 796] [29, 227, 426, 733, 797] [30, 226, 427, 734, 798] [31, 225, 424, 735, 799] [32, 228, 429, 736, 800] [33, 231, 430, 737, 801] [34, 230, 431, 738, 802] [35, 229, 428, 739, 803] [36, 232, 433, 740, 804] [37, 235, 434, 741, 805] [38, 234, 435, 742, 806] [39, 233, 432, 743, 807] [40, 236, 437, 744, 808] [41, 239, 438, 745, 809] [42, 238, 439, 746, 810] [43, 237, 436, 747, 811] [44, 240, 441, 748, 812] [45, 243, 442, 749, 813] [46, 242, 443, 750, 814] [47, 241, 440, 751, 815] [48, 244, 445, 752, 816] [49, 247, 446, 753, 817] [50, 246, 447, 754, 818] [51, 245, 444, 755, 819] [52, 248, 385, 756, 820] [53, 251, 386, 757, 821] [54, 250, 388, 758, 822] [55, 249, 384, 759, 823] [56, 252, 389, 760, 824] [57, 255, 390, 761, 825] [58, 254, 391, 762, 826] [59, 253, 387, 763, 827] [60, 192, 393, 764, 828] [61, 196, 394, 765, 829] [62, 194, 395, 766, 830] [63, 193, 392, 767, 831] [64, 259, 461, 704, 771] [65, 263, 462, 705, 773] [66, 262, 463, 706, 774] [67, 264, 465, 707, 776] [68, 261, 460, 708, 775] [69, 267, 466, 709, 777] [70, 266, 467, 710, 778] [71, 265, 464, 711, 779] [72, 268, 469, 712, 780] [73, 271, 470, 713, 781] [74, 270, 471, 714, 782] [75, 269, 468, 715, 783] [76, 272, 473, 716, 784] [77, 275, 474, 717, 785] [78, 274, 475, 718, 786] [79, 273, 472, 719, 787] [80, 276, 477, 720, 788] [81, 279, 478, 721, 789] [82, 278, 479, 722, 790] [83, 277, 476, 723, 791] [84, 280, 481, 724, 792] [85, 283, 482, 725, 793] [86, 282, 483, 726, 794] [87, 281, 480, 727, 795] [88, 284, 485, 728, 796] [89, 287, 486, 729, 797] [90, 286, 487, 730, 798] [91, 285, 484, 731, 799] [92, 288, 489, 732, 800] [93, 291, 490, 733, 801] [94, 290, 491, 734, 802] [95, 289, 488, 735, 803] [96, 292, 493, 736, 804] [97, 295, 494, 737, 805] [98, 294, 495, 738, 806] [99, 293, 492, 739, 807] [100, 296, 497, 740, 808] [101, 299, 498, 741, 809] [102, 298, 499, 742, 810] [103, 297, 496, 743, 811] [104, 300, 501, 744, 812] [105, 303, 502, 745, 813] [106, 302, 503, 746, 814] [107, 301, 500, 747, 815] [108, 304, 505, 748, 816] [109, 307, 506, 749, 817] [110, 306, 507, 750, 818] [111, 305, 504, 751, 819] [112, 308, 509, 752, 820] [113, 311, 510, 753, 821] [114, 310, 511, 754, 822] [115, 309, 508, 755, 823] [116, 312, 449, 756, 824] [117, 315, 450, 757, 825] [118, 314, 452, 758, 826] [119, 313, 448, 759, 827] [120, 316, 453, 760, 828] [121, 319, 454, 761, 829] [122, 318, 455, 762, 830] [123, 317, 451, 763, 831] [124, 256, 457, 764, 768] [125, 260, 458, 765, 769] [126, 258, 459, 766, 770] [128, 323, 525, 704, 793] [129, 327, 526, 705, 792] [130, 326, 527, 706, 795] [131, 328, 529, 707, 799] [132, 325, 524, 708, 794] [133, 331, 530, 709, 798] [134, 330, 531, 710, 797] [135, 329, 528, 711, 796] [136, 332, 533, 712, 801] [137, 335, 534, 713, 800] [138, 334, 535, 714, 803] [139, 333, 532, 715, 802] [140, 336, 537, 716, 807] [141, 339, 538, 717, 806] [142, 338, 539, 718, 805] [143, 337, 536, 719, 804] [144, 340, 541, 720, 809] [145, 343, 542, 721, 808] [146, 342, 543, 722, 811] [147, 341, 540, 723, 810] [148, 344, 545, 724, 815] [149, 347, 546, 725, 814] [150, 346, 547, 726, 813] [151, 345, 544, 727, 812] [152, 348, 549, 728, 817] [153, 351, 550, 729, 816] [154, 350, 551, 730, 819] [155, 349, 548, 731, 818] [156, 352, 553, 732, 823] [157, 355, 554, 733, 822] [158, 354, 555, 734, 821] [159, 353, 552, 735, 820] [160, 356, 557, 736, 825] [161, 359, 558, 737, 824] [162, 358, 559, 738, 827] [163, 357, 556, 739, 826] [164, 360, 561, 740, 831] [165, 363, 562, 741, 830] [166, 362, 563, 742, 829] [167, 361, 560, 743, 828] [168, 364, 565, 744, 769] [169, 367, 566, 745, 768] [170, 366, 567, 746, 772] [171, 365, 564, 747, 770] [172, 368, 569, 748, 775] [173, 371, 570, 749, 774] [174, 370, 571, 750, 773] [175, 369, 568, 751, 771] [176, 372, 573, 752, 777] [177, 375, 574, 753, 776] [178, 374, 575, 754, 779] [179, 373, 572, 755, 778] [180, 376, 513, 756, 783] [181, 379, 514, 757, 782] [182, 378, 516, 758, 781] [183, 377, 512, 759, 780] [184, 380, 517, 760, 785] [185, 383, 518, 761, 784] [186, 382, 519, 762, 787] [187, 381, 515, 763, 786] [188, 320, 521, 764, 791] [189, 324, 522, 765, 790] [190, 322, 523, 766, 789]
H_Z (382 checks, sparse supports)
[0, 64, 128, 576, 704] [1, 65, 129, 577, 705] [2, 66, 130, 578, 706] [3, 67, 131, 579, 707] [4, 68, 132, 580, 708] [5, 69, 133, 581, 709] [6, 70, 134, 582, 710] [7, 71, 135, 583, 711] [8, 72, 136, 584, 712] [9, 73, 137, 585, 713] [10, 74, 138, 586, 714] [11, 75, 139, 587, 715] [12, 76, 140, 588, 716] [13, 77, 141, 589, 717] [14, 78, 142, 590, 718] [15, 79, 143, 591, 719] [16, 80, 144, 592, 720] [17, 81, 145, 593, 721] [18, 82, 146, 594, 722] [19, 83, 147, 595, 723] [20, 84, 148, 596, 724] [21, 85, 149, 597, 725] [22, 86, 150, 598, 726] [23, 87, 151, 599, 727] [24, 88, 152, 600, 728] [25, 89, 153, 601, 729] [26, 90, 154, 602, 730] [27, 91, 155, 603, 731] [28, 92, 156, 604, 732] [29, 93, 157, 605, 733] [30, 94, 158, 606, 734] [31, 95, 159, 607, 735] [32, 96, 160, 608, 736] [33, 97, 161, 609, 737] [34, 98, 162, 610, 738] [35, 99, 163, 611, 739] [36, 100, 164, 612, 740] [37, 101, 165, 613, 741] [38, 102, 166, 614, 742] [39, 103, 167, 615, 743] [40, 104, 168, 616, 744] [41, 105, 169, 617, 745] [42, 106, 170, 618, 746] [43, 107, 171, 619, 747] [44, 108, 172, 620, 748] [45, 109, 173, 621, 749] [46, 110, 174, 622, 750] [47, 111, 175, 623, 751] [48, 112, 176, 624, 752] [49, 113, 177, 625, 753] [50, 114, 178, 626, 754] [51, 115, 179, 627, 755] [52, 116, 180, 628, 756] [53, 117, 181, 629, 757] [54, 118, 182, 630, 758] [55, 119, 183, 631, 759] [56, 120, 184, 632, 760] [57, 121, 185, 633, 761] [58, 122, 186, 634, 762] [59, 123, 187, 635, 763] [60, 124, 188, 636, 764] [61, 125, 189, 637, 765] [62, 126, 190, 638, 766] [63, 127, 191, 639, 767] [0, 124, 169, 640, 768] [1, 125, 168, 641, 769] [2, 126, 171, 642, 770] [3, 64, 175, 643, 771] [4, 127, 170, 644, 772] [5, 65, 174, 645, 773] [6, 66, 173, 646, 774] [7, 68, 172, 647, 775] [8, 67, 177, 648, 776] [9, 69, 176, 649, 777] [10, 70, 179, 650, 778] [11, 71, 178, 651, 779] [12, 72, 183, 652, 780] [13, 73, 182, 653, 781] [14, 74, 181, 654, 782] [15, 75, 180, 655, 783] [16, 76, 185, 656, 784] [17, 77, 184, 657, 785] [18, 78, 187, 658, 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Code ID 832-68-10-group64 · download JSON · raw on GitHub