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[[780,64,10]] d ≤
n
780
k
64
d
10
kd²/n
8.205
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)
[1, 19, 33, 44, 48, 61, 79, 93, 104, 108]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[0, 16, 32, 45, 48, 180, 196, 212, 225, 228]
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 2.295) · H_Z 1–3 (mean 2.295)
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): 536 (1,3): 237 (2,1): 15 (2,2): 1087 (2,3): 2125 (2,4): 353 (3,1): 33 (3,2): 2256 (3,3): 10643 (3,4): 6719 (3,5): 2828 (3,6): 1053
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): 536 (1,3): 237 (2,1): 15 (2,2): 1087 (2,3): 2125 (2,4): 353 (3,1): 33 (3,2): 2256 (3,3): 10643 (3,4): 6719 (3,5): 2828 (3,6): 1053

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction G=A5, even permutations of(0,1,2,3,4) in lexicographic order, composition a*b sends i to a[b[i]]. Set row-action L_g[t,u]=1 iff u=g*t, R_h[t,u]=1 iff u=t*h. A=[[I,I,I],[I,L_16,L_26]], B=[[I,I,I],[I,R_17,R_25]]. X(i,j) has A(i,k) on VV(k,j) and B(l,j) on CC(i,l); Z(k,l) has transpose(B(l,j)) on VV(k,j) and transpose(A(i,k)) on CC(i,l). Keep the lexicographically first independent rows in each sector. Full matrices have360 rows per sector, ranks358 each.
model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)
builds on https://arxiv.org/abs/2111.03654, https://arxiv.org/abs/2607.28795, https://arxiv.org/abs/2607.27644, https://arxiv.org/abs/1611.07164
date 2026-10-17
notes Displayed presentation: maximum check weight 5, maximum qubit degree 6, 716 checks and total incidence 3580.
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

[[780,64,10]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

G=A5, even permutations of(0,1,2,3,4) in lexicographic order, composition a*b sends i to a[b[i]]. Set row-action L_g[t,u]=1 iff u=g*t, R_h[t,u]=1 iff u=t*h. A=[[I,I,I],[I,L_16,L_26]], B=[[I,I,I],[I,R_17,R_25]]. X(i,j) has A(i,k) on VV(k,j) and B(l,j) on CC(i,l); Z(k,l) has transpose(B(l,j)) on VV(k,j) and transpose(A(i,k)) on CC(i,l). Keep the lexicographically first independent rows in each sector. Full matrices have360 rows per sector, ranks358 each. The nine VV(k,j) blocks use index 3*k+j, followed by four CC(i,l) blocks with index 9+2*i+l. Each block has 60 qubits; block b, group element t has label 60*b+t.

The displayed presentation has maximum check weight 5, maximum qubit degree 6, 716 checks and total incidence 3580.

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/780-64-10-a5.json. Allocate zero binary matrices H_X and H_Z with 780 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://arxiv.org/abs/2111.03654
  • https://arxiv.org/abs/2607.28795
  • https://arxiv.org/abs/2607.27644
  • https://arxiv.org/abs/1611.07164

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

Parity checks

X-checks 358 (max weight 5) · Z-checks 358 (max weight 5)
H_X (358 checks, sparse supports)
[0, 180, 360, 540, 600] [1, 181, 361, 541, 601] [2, 182, 362, 542, 602] [3, 183, 363, 543, 603] [4, 184, 364, 544, 604] [5, 185, 365, 545, 605] [6, 186, 366, 546, 606] [7, 187, 367, 547, 607] [8, 188, 368, 548, 608] [9, 189, 369, 549, 609] [10, 190, 370, 550, 610] [11, 191, 371, 551, 611] [12, 192, 372, 552, 612] [13, 193, 373, 553, 613] [14, 194, 374, 554, 614] [15, 195, 375, 555, 615] [16, 196, 376, 556, 616] [17, 197, 377, 557, 617] [18, 198, 378, 558, 618] [19, 199, 379, 559, 619] [20, 200, 380, 560, 620] [21, 201, 381, 561, 621] [22, 202, 382, 562, 622] [23, 203, 383, 563, 623] [24, 204, 384, 564, 624] [25, 205, 385, 565, 625] [26, 206, 386, 566, 626] [27, 207, 387, 567, 627] [28, 208, 388, 568, 628] [29, 209, 389, 569, 629] [30, 210, 390, 570, 630] [31, 211, 391, 571, 631] [32, 212, 392, 572, 632] [33, 213, 393, 573, 633] [34, 214, 394, 574, 634] [35, 215, 395, 575, 635] [36, 216, 396, 576, 636] [37, 217, 397, 577, 637] [38, 218, 398, 578, 638] [39, 219, 399, 579, 639] [40, 220, 400, 580, 640] [41, 221, 401, 581, 641] [42, 222, 402, 582, 642] [43, 223, 403, 583, 643] [44, 224, 404, 584, 644] [45, 225, 405, 585, 645] [46, 226, 406, 586, 646] [47, 227, 407, 587, 647] [48, 228, 408, 588, 648] [49, 229, 409, 589, 649] [50, 230, 410, 590, 650] [51, 231, 411, 591, 651] [52, 232, 412, 592, 652] [53, 233, 413, 593, 653] [54, 234, 414, 594, 654] [55, 235, 415, 595, 655] [56, 236, 416, 596, 656] [57, 237, 417, 597, 657] [58, 238, 418, 598, 658] [59, 239, 419, 599, 659] [60, 240, 420, 540, 617] [61, 241, 421, 541, 619] [62, 242, 422, 542, 623] [63, 243, 423, 543, 628] [64, 244, 424, 544, 632] [65, 245, 425, 545, 634] [66, 246, 426, 546, 641] [67, 247, 427, 547, 643] [68, 248, 428, 548, 647] [69, 249, 429, 549, 652] [70, 250, 430, 550, 656] [71, 251, 431, 551, 658] [72, 252, 432, 552, 604] [73, 253, 433, 553, 608] [74, 254, 434, 554, 610] [75, 255, 435, 555, 626] [76, 256, 436, 556, 630] [77, 257, 437, 557, 635] [78, 258, 438, 558, 637] [79, 259, 439, 559, 644] [80, 260, 440, 560, 645] [81, 261, 441, 561, 650] [82, 262, 442, 562, 654] [83, 263, 443, 563, 659] [84, 264, 444, 564, 602] [85, 265, 445, 565, 606] [86, 266, 446, 566, 611] [87, 267, 447, 567, 613] [88, 268, 448, 568, 620] [89, 269, 449, 569, 621] [90, 270, 450, 570, 638] [91, 271, 451, 571, 639] [92, 272, 452, 572, 646] [93, 273, 453, 573, 648] [94, 274, 454, 574, 653] [95, 275, 455, 575, 657] [96, 276, 456, 576, 600] [97, 277, 457, 577, 605] [98, 278, 458, 578, 609] [99, 279, 459, 579, 614] [100, 280, 460, 580, 615] [101, 281, 461, 581, 622] [102, 282, 462, 582, 624] [103, 283, 463, 583, 629] [104, 284, 464, 584, 633] [105, 285, 465, 585, 649] [106, 286, 466, 586, 651] [107, 287, 467, 587, 655] [108, 288, 468, 588, 601] [109, 289, 469, 589, 603] [110, 290, 470, 590, 607] [111, 291, 471, 591, 612] [112, 292, 472, 592, 616] [113, 293, 473, 593, 618] [114, 294, 474, 594, 625] [115, 295, 475, 595, 627] [116, 296, 476, 596, 631] [117, 297, 477, 597, 636] [118, 298, 478, 598, 640] [119, 299, 479, 599, 642] [120, 300, 480, 540, 625] [121, 301, 481, 541, 638] [122, 302, 482, 542, 649] [123, 303, 483, 543, 614] [124, 304, 484, 544, 636] [125, 305, 485, 545, 650] [126, 306, 486, 546, 612] [127, 307, 487, 547, 626] [128, 308, 488, 548, 648] [129, 309, 489, 549, 613] [130, 310, 490, 550, 624] [131, 311, 491, 551, 637] [132, 312, 492, 552, 629] [133, 313, 493, 553, 640] [134, 314, 494, 554, 653] [135, 315, 495, 555, 601] [136, 316, 496, 556, 641] [137, 317, 497, 557, 651] [138, 318, 498, 558, 602] [139, 319, 499, 559, 627] [140, 320, 500, 560, 652] [141, 321, 501, 561, 600] [142, 322, 502, 562, 628] [143, 323, 503, 563, 639] [144, 324, 504, 564, 616] [145, 325, 505, 565, 644] [146, 326, 506, 566, 655] [147, 327, 507, 567, 605] [148, 328, 508, 568, 642] [149, 329, 509, 569, 656] [150, 330, 510, 570, 603] [151, 331, 511, 571, 617] [152, 332, 512, 572, 654] [153, 333, 513, 573, 604] [154, 334, 514, 574, 615] [155, 335, 515, 575, 643] [156, 336, 516, 576, 620] [157, 337, 517, 577, 631] [158, 338, 518, 578, 659] [159, 339, 519, 579, 607] [160, 340, 520, 580, 632] [161, 341, 521, 581, 657] [162, 342, 522, 582, 608] [163, 343, 523, 583, 618] [164, 344, 524, 584, 658] [165, 345, 525, 585, 606] [166, 346, 526, 586, 619] [167, 347, 527, 587, 630] [168, 348, 528, 588, 622] [169, 349, 529, 589, 635] [170, 350, 530, 590, 646] [171, 351, 531, 591, 611] [172, 352, 532, 592, 633] [173, 353, 533, 593, 647] [174, 354, 534, 594, 609] [175, 355, 535, 595, 623] [176, 356, 536, 596, 645] [177, 357, 537, 597, 610] [178, 358, 538, 598, 621] [179, 359, 539, 599, 634] [0, 196, 386, 660, 720] [1, 197, 384, 661, 721] [2, 195, 385, 662, 722] [3, 199, 393, 663, 723] [4, 200, 394, 664, 724] [5, 198, 395, 665, 725] [6, 202, 387, 666, 726] [7, 203, 389, 667, 727] [8, 201, 388, 668, 728] [9, 192, 390, 669, 729] [10, 193, 392, 670, 730] [11, 194, 391, 671, 731] [12, 208, 365, 672, 732] [13, 209, 363, 673, 733] [14, 207, 364, 674, 734] [15, 211, 370, 675, 735] [16, 212, 369, 676, 736] [17, 210, 371, 677, 737] [18, 214, 360, 678, 738] [19, 215, 362, 679, 739] [20, 213, 361, 680, 740] [21, 204, 367, 681, 741] [22, 205, 368, 682, 742] [23, 206, 366, 683, 743] [24, 220, 414, 684, 744] [25, 221, 415, 685, 745] [26, 219, 416, 686, 746] [27, 223, 409, 687, 747] [28, 224, 408, 688, 748] [29, 222, 410, 689, 749] [30, 226, 412, 690, 750] [31, 227, 411, 691, 751] [32, 225, 413, 692, 752] [33, 216, 419, 693, 753] [34, 217, 417, 694, 754] [35, 218, 418, 695, 755] [36, 232, 375, 696, 756] [37, 233, 377, 697, 757] [38, 231, 376, 698, 758] [39, 235, 372, 699, 759] [40, 236, 374, 700, 760] [41, 234, 373, 701, 761] [42, 238, 382, 702, 762] [43, 239, 381, 703, 763] [44, 237, 383, 704, 764] [45, 228, 379, 705, 765] [46, 229, 378, 706, 766] [47, 230, 380, 707, 767] [48, 180, 402, 708, 768] [49, 181, 404, 709, 769] [50, 182, 403, 710, 770] [51, 183, 397, 711, 771] [52, 184, 398, 712, 772] [53, 185, 396, 713, 773] [54, 186, 407, 714, 774] [55, 187, 405, 715, 775] [56, 188, 406, 716, 776] [57, 189, 400, 717, 777] [58, 190, 399, 718, 778] [59, 191, 401, 719, 779] [60, 256, 446, 660, 737] [61, 257, 444, 661, 739] [62, 255, 445, 662, 743] [63, 259, 453, 663, 748] [64, 260, 454, 664, 752] [65, 258, 455, 665, 754] [66, 262, 447, 666, 761] [67, 263, 449, 667, 763] [68, 261, 448, 668, 767] [69, 252, 450, 669, 772] [70, 253, 452, 670, 776] [71, 254, 451, 671, 778] [72, 268, 425, 672, 724] [73, 269, 423, 673, 728] [74, 267, 424, 674, 730] [75, 271, 430, 675, 746] [76, 272, 429, 676, 750] [77, 270, 431, 677, 755] [78, 274, 420, 678, 757] [79, 275, 422, 679, 764] [80, 273, 421, 680, 765] [81, 264, 427, 681, 770] [82, 265, 428, 682, 774] [83, 266, 426, 683, 779] [84, 280, 474, 684, 722] [85, 281, 475, 685, 726] [86, 279, 476, 686, 731] [87, 283, 469, 687, 733] [88, 284, 468, 688, 740] [89, 282, 470, 689, 741] [90, 286, 472, 690, 758] [91, 287, 471, 691, 759] [92, 285, 473, 692, 766] [93, 276, 479, 693, 768] [94, 277, 477, 694, 773] [95, 278, 478, 695, 777] [96, 292, 435, 696, 720] [97, 293, 437, 697, 725] [98, 291, 436, 698, 729] [99, 295, 432, 699, 734] [100, 296, 434, 700, 735] [101, 294, 433, 701, 742] [102, 298, 442, 702, 744] [103, 299, 441, 703, 749] [104, 297, 443, 704, 753] [105, 288, 439, 705, 769] [106, 289, 438, 706, 771] [107, 290, 440, 707, 775] [108, 240, 462, 708, 721] [109, 241, 464, 709, 723] [110, 242, 463, 710, 727] [111, 243, 457, 711, 732] [112, 244, 458, 712, 736] [113, 245, 456, 713, 738] [114, 246, 467, 714, 745] [115, 247, 465, 715, 747] [116, 248, 466, 716, 751] [117, 249, 460, 717, 756] [118, 250, 459, 718, 760] [120, 316, 506, 660, 745] [121, 317, 504, 661, 758] [122, 315, 505, 662, 769] [123, 319, 513, 663, 734] [124, 320, 514, 664, 756] [125, 318, 515, 665, 770] [126, 322, 507, 666, 732] [127, 323, 509, 667, 746] [128, 321, 508, 668, 768] [129, 312, 510, 669, 733] [130, 313, 512, 670, 744] [131, 314, 511, 671, 757] [132, 328, 485, 672, 749] [133, 329, 483, 673, 760] [134, 327, 484, 674, 773] [135, 331, 490, 675, 721] [136, 332, 489, 676, 761] [137, 330, 491, 677, 771] [138, 334, 480, 678, 722] [139, 335, 482, 679, 747] [140, 333, 481, 680, 772] [141, 324, 487, 681, 720] [142, 325, 488, 682, 748] [143, 326, 486, 683, 759] [144, 340, 534, 684, 736] [145, 341, 535, 685, 764] [146, 339, 536, 686, 775] [147, 343, 529, 687, 725] [148, 344, 528, 688, 762] [149, 342, 530, 689, 776] [150, 346, 532, 690, 723] [151, 347, 531, 691, 737] [152, 345, 533, 692, 774] [153, 336, 539, 693, 724] [154, 337, 537, 694, 735] [155, 338, 538, 695, 763] [156, 352, 495, 696, 740] [157, 353, 497, 697, 751] [158, 351, 496, 698, 779] [159, 355, 492, 699, 727] [160, 356, 494, 700, 752] [161, 354, 493, 701, 777] [162, 358, 502, 702, 728] [163, 359, 501, 703, 738] [164, 357, 503, 704, 778] [165, 348, 499, 705, 726] [166, 349, 498, 706, 739] [167, 350, 500, 707, 750] [168, 300, 522, 708, 742] [169, 301, 524, 709, 755] [170, 302, 523, 710, 766] [171, 303, 517, 711, 731] [172, 304, 518, 712, 753] [173, 305, 516, 713, 767] [174, 306, 527, 714, 729] [175, 307, 525, 715, 743] [176, 308, 526, 716, 765] [177, 309, 520, 717, 730] [178, 310, 519, 718, 741]
H_Z (358 checks, sparse supports)
[0, 60, 120, 540, 660] [1, 61, 121, 541, 661] [2, 62, 122, 542, 662] [3, 63, 123, 543, 663] [4, 64, 124, 544, 664] [5, 65, 125, 545, 665] [6, 66, 126, 546, 666] [7, 67, 127, 547, 667] [8, 68, 128, 548, 668] [9, 69, 129, 549, 669] [10, 70, 130, 550, 670] [11, 71, 131, 551, 671] [12, 72, 132, 552, 672] [13, 73, 133, 553, 673] [14, 74, 134, 554, 674] [15, 75, 135, 555, 675] [16, 76, 136, 556, 676] [17, 77, 137, 557, 677] [18, 78, 138, 558, 678] [19, 79, 139, 559, 679] [20, 80, 140, 560, 680] [21, 81, 141, 561, 681] [22, 82, 142, 562, 682] [23, 83, 143, 563, 683] [24, 84, 144, 564, 684] [25, 85, 145, 565, 685] [26, 86, 146, 566, 686] [27, 87, 147, 567, 687] [28, 88, 148, 568, 688] [29, 89, 149, 569, 689] [30, 90, 150, 570, 690] [31, 91, 151, 571, 691] [32, 92, 152, 572, 692] [33, 93, 153, 573, 693] [34, 94, 154, 574, 694] [35, 95, 155, 575, 695] [36, 96, 156, 576, 696] [37, 97, 157, 577, 697] [38, 98, 158, 578, 698] [39, 99, 159, 579, 699] [40, 100, 160, 580, 700] [41, 101, 161, 581, 701] [42, 102, 162, 582, 702] [43, 103, 163, 583, 703] [44, 104, 164, 584, 704] [45, 105, 165, 585, 705] [46, 106, 166, 586, 706] [47, 107, 167, 587, 707] [48, 108, 168, 588, 708] [49, 109, 169, 589, 709] [50, 110, 170, 590, 710] [51, 111, 171, 591, 711] [52, 112, 172, 592, 712] [53, 113, 173, 593, 713] [54, 114, 174, 594, 714] [55, 115, 175, 595, 715] [56, 116, 176, 596, 716] [57, 117, 177, 597, 717] [58, 118, 178, 598, 718] [59, 119, 179, 599, 719] [0, 96, 141, 600, 720] [1, 108, 135, 601, 721] [2, 84, 138, 602, 722] [3, 109, 150, 603, 723] [4, 72, 153, 604, 724] [5, 97, 147, 605, 725] [6, 85, 165, 606, 726] [7, 110, 159, 607, 727] [8, 73, 162, 608, 728] [9, 98, 174, 609, 729] [10, 74, 177, 610, 730] [11, 86, 171, 611, 731] [12, 111, 126, 612, 732] [13, 87, 129, 613, 733] [14, 99, 123, 614, 734] [15, 100, 154, 615, 735] [16, 112, 144, 616, 736] [17, 60, 151, 617, 737] [18, 113, 163, 618, 738] [19, 61, 166, 619, 739] [20, 88, 156, 620, 740] [21, 89, 178, 621, 741] [22, 101, 168, 622, 742] [23, 62, 175, 623, 743] [24, 102, 130, 624, 744] [25, 114, 120, 625, 745] [26, 75, 127, 626, 746] [27, 115, 139, 627, 747] [28, 63, 142, 628, 748] [29, 103, 132, 629, 749] [30, 76, 167, 630, 750] [31, 116, 157, 631, 751] [32, 64, 160, 632, 752] [33, 104, 172, 633, 753] [34, 65, 179, 634, 754] [35, 77, 169, 635, 755] [36, 117, 124, 636, 756] [37, 78, 131, 637, 757] [38, 90, 121, 638, 758] [39, 91, 143, 639, 759] [40, 118, 133, 640, 760] [41, 66, 136, 641, 761] [42, 119, 148, 642, 762] [43, 67, 155, 643, 763] [44, 79, 145, 644, 764] [45, 80, 176, 645, 765] [46, 92, 170, 646, 766] [47, 68, 173, 647, 767] [48, 93, 128, 648, 768] [49, 105, 122, 649, 769] [50, 81, 125, 650, 770] [51, 106, 137, 651, 771] [52, 69, 140, 652, 772] [53, 94, 134, 653, 773] [54, 82, 152, 654, 774] [55, 107, 146, 655, 775] [56, 70, 149, 656, 776] [57, 95, 161, 657, 777] [58, 71, 164, 658, 778] [59, 83, 158, 659, 779] [180, 240, 300, 540, 708] [181, 241, 301, 541, 709] [182, 242, 302, 542, 710] [183, 243, 303, 543, 711] [184, 244, 304, 544, 712] [185, 245, 305, 545, 713] [186, 246, 306, 546, 714] [187, 247, 307, 547, 715] [188, 248, 308, 548, 716] [189, 249, 309, 549, 717] [190, 250, 310, 550, 718] [191, 251, 311, 551, 719] [192, 252, 312, 552, 669] [193, 253, 313, 553, 670] [194, 254, 314, 554, 671] [195, 255, 315, 555, 662] [196, 256, 316, 556, 660] [197, 257, 317, 557, 661] [198, 258, 318, 558, 665] [199, 259, 319, 559, 663] [200, 260, 320, 560, 664] [201, 261, 321, 561, 668] [202, 262, 322, 562, 666] [203, 263, 323, 563, 667] [204, 264, 324, 564, 681] [205, 265, 325, 565, 682] [206, 266, 326, 566, 683] [207, 267, 327, 567, 674] [208, 268, 328, 568, 672] [209, 269, 329, 569, 673] [210, 270, 330, 570, 677] [211, 271, 331, 571, 675] [212, 272, 332, 572, 676] [213, 273, 333, 573, 680] [214, 274, 334, 574, 678] [215, 275, 335, 575, 679] [216, 276, 336, 576, 693] [217, 277, 337, 577, 694] [218, 278, 338, 578, 695] [219, 279, 339, 579, 686] [220, 280, 340, 580, 684] [221, 281, 341, 581, 685] [222, 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Code ID 780-64-10-a5 · download JSON · raw on GitHub