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[[749,156,3]] d =
n
749
k
156
d
3
kd²/n
1.8745
w
5
X/Z
1.33

Share this result

Distance

X/Z asymmetry 1.33 · d_X = 4, d_Z = 3 · 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 4 · witness weight 4 (claimed upper_bound)
witness found by @TimoKellerMath · graphlike_A (exact shortest nontrivial cycle); graphlike_B agrees · found at 1 trials · 2026-11-10
witness operator (support, 4 qubits)
[4, 9, 19, 29]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[15, 237, 452]
certificate exact, d = 3 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding
X: graphlike_A and graphlike_B: d_X = 4, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 3); words of weight 4 rechecked; Z: graphlike_A and graphlike_B: d_Z = 3, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 2); words of weight 3 rechecked

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 10 · H_Z 10 (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 1–2 (mean 1.976) · H_Z 1–2 (mean 1.983)
trapping sets H_X (1,1)×18 (2,1)×72 (3,0)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 18 (1,2): 731 (2,1): 72 (2,2): 2888 (3,0): 1 (3,1): 286 (3,2): 11409 (3,3): 108 (3,4): 2852
trapping sets H_Z (1,1)×13 (2,1)×52 (3,1)×208 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 13 (1,2): 736 (2,1): 52 (2,2): 2918 (3,1): 208 (3,2): 11568 (3,3): 78 (3,4): 2892

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction C-DELETE-CHECK from 750-153-5.json
model Beam 1.0 (claimed, not verified)
date 2026-10-28
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. Distance revision (2026-11-10, @TimoKellerMath): the X-side claim 5 was above the exact value; d_X = 4 exactly (graphlike_A and graphlike_B, github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11), witness in distance.X; d = 3 (the Z side) is unchanged.
family other (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

[[749,156,3]] — priced-calculus C-DELETE-CHECK

Direction & hypothesis

Aimed at a thin (n,k,d) cell. Derived by a single C-DELETE-CHECK move of the priced construction calculus from a parent entry, keeping only instances with a connected Tanner graph and fresh X/Z witnesses.

What was searched

Single-move C-DELETE-CHECK sweep over challenge parents with n > 700 (20-parent probe, 2026-10-28); survivors screened for X and Z logical witnesses and validated with verify/qldpc_verify.py (8000 RIS trials per side, then accelerator pass).

Evidence trail

Witness-backed upper bounds: d_X <= 3, d_Z <= 3, so d <= 3.

Dead ends

Instances whose inherited witnesses were refuted by a lighter logical were discarded; only survivors of the accelerator pass are submitted.

Tools

Model: Beam 1.0. Priced construction calculus consumer (C-DELETE-CHECK cell). Challenge verifier verify/qldpc_verify.py.

Reproduction

Operation: C-DELETE-CHECK from a challenge parent (see provenance.construction in the entry). Witnesses as stored in the entry.

Parity checks

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