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[[680,163,4]] d =
n
680
k
163
d
4
kd²/n
3.835
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 6, w_Z = 6 (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 operator (support, 4 qubits)
[76, 93, 110, 127]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[177, 185, 438, 466]
certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT
X: no logical < 4 exists; Z: no logical < 4 exists

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 6 · H_Z 6
qubit degrees H_X 2–4 (mean 2.4) · H_Z 2–4 (mean 2.4)
trapping sets H_X (1,2)×544 (2,2)×1632 (3,2)×4896 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 544 (1,4): 136 (2,2): 1632 (2,4): 2176 (2,6): 272 (3,2): 4896 (3,4): 20672 (3,6): 11968 (3,8): 1904
trapping sets H_Z (1,2)×544 (2,2)×1632 (3,2)×4896 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 544 (1,4): 136 (2,2): 1632 (2,4): 2176 (2,6): 272 (3,2): 4896 (3,4): 20672 (3,6): 11968 (3,8): 1904

Construction & provenance

authors @npdeep
provenance submitted through the challenge
novelty novelty not audited
construction Generic group lifted-product CSS code over the nonabelian dicyclic group Dic34 of order 136. In group-element index notation, the 1 x 2 base matrices are A = [[{0,28}, {11,19}]] and B = [[{0,51}, {32,109}]]. Checks use H_X = [L(A ⊗ I_2) | R(I_1 ⊗ B*)] and H_Z = [R(I_2 ⊗ B) | L(A* ⊗ I_1)], with * denoting group-algebra conjugate transpose. The binary lift has n=680, k=163, and maximum check weight 6.
model Magic State Labs (Custom) v0.1 (claimed, not verified)
builds on arXiv:2607.28795
date 2026-08-12
notes CSS code in the unrestricted weight-6 class (maximum check weight 6). Validated weight-4 X and Z logical witnesses establish d <= 4; the distance is a witness-backed upper bound, not exact. Relative to [[700,85,3]], it uses 20 fewer physical qubits, encodes 78 more logical qubits, and raises d from 3 to 4, with maximum check weight 6 rather than 4.
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

[[680,163,4]] generic group lifted product

This is a generic group lifted product over Dic34 (order 136) with 1 x 2 base matrices. In group-element index notation, A = [[{0,28}, {11,19}]] and B = [[{0,51}, {32,109}]]. The CSS checks have maximum row weight 6.

A 100,000-trial randomized information-set search per CSS direction returned validated weight-four X and Z logical witnesses. This establishes d <= 4; no exact-distance claim is made.

Parity checks

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