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[[900,2,30]] d ≤
n
900
k
2
d
30
kd²/n
2.0
w
4
X/Z
1
g
0.125
r
2.8284
layers
1
swaps
2464

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Distance

X/Z asymmetry 1 · d_X ≤ 30, d_Z ≤ 30 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 30 · witness weight 30 (claimed upper_bound)
witness operator (support, 30 qubits)
[171, 172, 173, 174, 175, 176, 177, 178, 200, 209, 210, 211, 212, 229, 243, 258, 274, 287, 305, 316, 336, 345, 367, 374, 398, 399, 400, 403, 431, 432]
d_Z 30 · witness weight 30 (claimed upper_bound)
witness operator (support, 30 qubits)
[9, 39, 70, 101, 132, 162, 192, 222, 251, 281, 311, 340, 369, 398, 428, 458, 487, 517, 547, 577, 608, 639, 670, 700, 729, 759, 789, 819, 850, 880]
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 4 · H_Z 4
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×900 (2,2)×2700 (3,2)×8100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 900 (2,2): 2700 (3,2): 8100 (3,4): 1800
trapping sets H_Z (1,2)×900 (2,2)×2700 (3,2)×8100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 900 (2,2): 2700 (3,2): 8100 (3,4): 1800
witness diameter X 30.0832 · Z 29.0689 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2.828
X checkZ checkqubit site (900)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 2464 nearest-neighbor SWAPs per round in total, at most 3 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Rotated toric code on the periodic 30x30 lattice: all plaquette checks, (i+j) even -> X; distance 30 (cycle lengths).
builds on arXiv:quant-ph/9707021
date 2026-09-20
notes Rotated surface/toric ladder fill: standard topological configuration, distance exact by construction (a row or column of plaquettes is a weight-d logical; none lighter exists). Staged for review; novelty vs the wider literature unverified beyond the cited family origin.
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

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

[[900,2,30]] rotated toric code (weight-4, 2D-local single-layer)

Direction & hypothesis

The board's least populated primary-track cell is weight-4 x local-2d-single (175 of 1,126 entries at the time of writing): the surface-code-like regime that most qLDPC families (weight >= 6 checks) cannot reach. The standard topological ladders are only seeded at a few sizes on the board (rotated surface [[L^2,1,L]] at L = 5,7,9,11; rotated toric [[L^2,2,L]] at L = 4,6,8), so the hypothesis was that the missing square sizes extend this cell's Pareto frontier on d, at both k = 1 (surface) and k = 2 (toric).

What was searched

Full enumeration of both rotated ladders under the board's admission caps (n <= 700, or the extended tier n <= 1000 with w <= 8 and d <= 40; w = 4 throughout): every square L with admissible(n, 4, L) true, plus every rectangular (L1, L2) configuration -- 1,270 configurations in total. Each was packaged with the kit's own submit.make_submission (RIS witness search) and passed through the trusted gate verify/validate_candidate.py. The gate's novelty verdict split the enumeration exactly along a structural line: squares advance, rectangles never (each [[L1*L2,1,L2]] is strictly dominated by its own square [[L2^2,1,L2]]: same k, same d, same weight, strictly fewer qubits). Only board-advancing survivors were kept; this PR is one of them.

Evidence trail

  • Witness search (kit surrogate, RIS): <= 3,000 trials/side scaled with n;

    • the lightest logical found on each side has weight 30 (a row or column of plaquettes), matching the exact-by-construction distance.

  • Trusted gate (verify/validate_candidate.py): verify ok, distance not

    • refuted (independent fresh-seed RIS search found nothing lighter), not a board duplicate (fingerprint and WL signature checked against the whole board), and board-advancing in the weight-4 x local-2d-single cell.

  • Claim carried: witness-backed upper bound, d <= 30. The distance is exact

    • by construction (no logical lighter than a minimum non-contractible chain exists on the torus), but it is submitted as an upper bound; a maintainer can run verify/certify.py to upgrade it.

Dead ends

  • Rectangular rotated surface/toric codes: all dominated by their own square

    • (see above) -- generated and gate-checked, then discarded.

  • [[9,1,3]] (dominated by the seeded [[7,1,3]]) and [[8,2,2]] (dominated by

    • [[4,2,2]]): gate-flagged as not board-advancing, discarded.

  • Rotated toric L = 10,12,14: parameter sets already on the board

    • ([[100,2,10]] generalized-bicycle, [[144,2,12]] trivariate-bicycle, [[196,2,14]] bivariate-bicycle), skipped as parameter duplicates.

  • An initial torus layout used a ring fold with a seam gap of 3 grid units,

    • which pushed the measured interaction radius past the single-layer cap for L >= 10 (demoting those codes to local-2d-bilayer). The seeded entries' fold convention (seam gap 1) keeps every plaquette diameter at 2*sqrt(2) for every L; that is what this submission uses.

  • Unrotated surface [[2d^2-2d+1,1,d]] and 2D color codes were attempted for a

    • companion fill but their from-scratch constructions failed self-checks; deferred until built from published constructions.

Tools

GLM 5.3 Flash (Zed coding agent); repo kit only (research/kit/submit.py, verify/validate_candidate.py, qldpc_verify), numpy-only constructions. Constructions were verified bit-for-bit (check-support set equality) against the board's own seeded codes/25-1-5.json (L = 5,7,9) and codes/16-2-4.json (L = 4,6,8) before scaling to new sizes.

Reproduction

Rotated toric code [[L^2,2,L]], L = 30 (even):

  • qubits q = (i, j) -> q = i*L + j on an L x L torus (periodic in both axes);
  • for each (i, j) in {0..L-1}^2 the periodic 2x2 plaquette

    • {(i,j), (i,j+1 mod L), (i+1 mod L, j), (i+1 mod L, j+1 mod L)} is an X check when i+j is even and a Z check when i+j is odd;

  • layout: ring fold per axis, i -> 2i for i < L/2 and i -> 2(L-i)-1 for

    • i >= L/2 (the seeded entries' convention; seam gap 1, so every plaquette has diameter 2*sqrt(2)), single layer. Distance: a non-contractible row or column of plaquettes is a weight-L logical; none lighter exists (cycle lengths of the torus).

Parity checks

X-checks 450 (max weight 4) · Z-checks 450 (max weight 4)
H_X (450 checks, sparse supports)
[0, 1, 30, 31] [2, 3, 32, 33] [4, 5, 34, 35] [6, 7, 36, 37] [8, 9, 38, 39] [10, 11, 40, 41] [12, 13, 42, 43] [14, 15, 44, 45] [16, 17, 46, 47] [18, 19, 48, 49] [20, 21, 50, 51] [22, 23, 52, 53] [24, 25, 54, 55] [26, 27, 56, 57] [28, 29, 58, 59] [31, 32, 61, 62] [33, 34, 63, 64] [35, 36, 65, 66] [37, 38, 67, 68] [39, 40, 69, 70] [41, 42, 71, 72] [43, 44, 73, 74] [45, 46, 75, 76] [47, 48, 77, 78] [49, 50, 79, 80] [51, 52, 81, 82] [53, 54, 83, 84] [55, 56, 85, 86] [57, 58, 87, 88] [30, 59, 60, 89] [60, 61, 90, 91] [62, 63, 92, 93] [64, 65, 94, 95] [66, 67, 96, 97] [68, 69, 98, 99] [70, 71, 100, 101] [72, 73, 102, 103] [74, 75, 104, 105] [76, 77, 106, 107] [78, 79, 108, 109] [80, 81, 110, 111] [82, 83, 112, 113] [84, 85, 114, 115] [86, 87, 116, 117] [88, 89, 118, 119] [91, 92, 121, 122] [93, 94, 123, 124] [95, 96, 125, 126] [97, 98, 127, 128] [99, 100, 129, 130] [101, 102, 131, 132] [103, 104, 133, 134] [105, 106, 135, 136] [107, 108, 137, 138] [109, 110, 139, 140] [111, 112, 141, 142] [113, 114, 143, 144] [115, 116, 145, 146] [117, 118, 147, 148] [90, 119, 120, 149] [120, 121, 150, 151] [122, 123, 152, 153] [124, 125, 154, 155] [126, 127, 156, 157] [128, 129, 158, 159] [130, 131, 160, 161] [132, 133, 162, 163] [134, 135, 164, 165] [136, 137, 166, 167] [138, 139, 168, 169] [140, 141, 170, 171] [142, 143, 172, 173] [144, 145, 174, 175] [146, 147, 176, 177] [148, 149, 178, 179] [151, 152, 181, 182] [153, 154, 183, 184] [155, 156, 185, 186] [157, 158, 187, 188] [159, 160, 189, 190] [161, 162, 191, 192] [163, 164, 193, 194] [165, 166, 195, 196] [167, 168, 197, 198] [169, 170, 199, 200] [171, 172, 201, 202] [173, 174, 203, 204] [175, 176, 205, 206] [177, 178, 207, 208] [150, 179, 180, 209] [180, 181, 210, 211] [182, 183, 212, 213] [184, 185, 214, 215] [186, 187, 216, 217] [188, 189, 218, 219] [190, 191, 220, 221] [192, 193, 222, 223] [194, 195, 224, 225] [196, 197, 226, 227] [198, 199, 228, 229] [200, 201, 230, 231] [202, 203, 232, 233] [204, 205, 234, 235] [206, 207, 236, 237] [208, 209, 238, 239] [211, 212, 241, 242] [213, 214, 243, 244] [215, 216, 245, 246] [217, 218, 247, 248] [219, 220, 249, 250] [221, 222, 251, 252] [223, 224, 253, 254] [225, 226, 255, 256] [227, 228, 257, 258] [229, 230, 259, 260] [231, 232, 261, 262] [233, 234, 263, 264] [235, 236, 265, 266] [237, 238, 267, 268] [210, 239, 240, 269] [240, 241, 270, 271] [242, 243, 272, 273] [244, 245, 274, 275] [246, 247, 276, 277] [248, 249, 278, 279] [250, 251, 280, 281] [252, 253, 282, 283] [254, 255, 284, 285] [256, 257, 286, 287] [258, 259, 288, 289] [260, 261, 290, 291] [262, 263, 292, 293] [264, 265, 294, 295] [266, 267, 296, 297] [268, 269, 298, 299] [271, 272, 301, 302] [273, 274, 303, 304] [275, 276, 305, 306] [277, 278, 307, 308] [279, 280, 309, 310] [281, 282, 311, 312] [283, 284, 313, 314] [285, 286, 315, 316] [287, 288, 317, 318] [289, 290, 319, 320] [291, 292, 321, 322] [293, 294, 323, 324] [295, 296, 325, 326] [297, 298, 327, 328] [270, 299, 300, 329] [300, 301, 330, 331] [302, 303, 332, 333] [304, 305, 334, 335] [306, 307, 336, 337] [308, 309, 338, 339] [310, 311, 340, 341] [312, 313, 342, 343] [314, 315, 344, 345] [316, 317, 346, 347] [318, 319, 348, 349] [320, 321, 350, 351] [322, 323, 352, 353] [324, 325, 354, 355] [326, 327, 356, 357] [328, 329, 358, 359] [331, 332, 361, 362] [333, 334, 363, 364] [335, 336, 365, 366] [337, 338, 367, 368] [339, 340, 369, 370] [341, 342, 371, 372] [343, 344, 373, 374] [345, 346, 375, 376] [347, 348, 377, 378] [349, 350, 379, 380] [351, 352, 381, 382] [353, 354, 383, 384] [355, 356, 385, 386] [357, 358, 387, 388] [330, 359, 360, 389] [360, 361, 390, 391] [362, 363, 392, 393] [364, 365, 394, 395] [366, 367, 396, 397] [368, 369, 398, 399] [370, 371, 400, 401] [372, 373, 402, 403] [374, 375, 404, 405] [376, 377, 406, 407] [378, 379, 408, 409] [380, 381, 410, 411] [382, 383, 412, 413] [384, 385, 414, 415] [386, 387, 416, 417] [388, 389, 418, 419] [391, 392, 421, 422] [393, 394, 423, 424] [395, 396, 425, 426] [397, 398, 427, 428] [399, 400, 429, 430] [401, 402, 431, 432] [403, 404, 433, 434] [405, 406, 435, 436] [407, 408, 437, 438] [409, 410, 439, 440] [411, 412, 441, 442] [413, 414, 443, 444] [415, 416, 445, 446] [417, 418, 447, 448] [390, 419, 420, 449] [420, 421, 450, 451] [422, 423, 452, 453] [424, 425, 454, 455] [426, 427, 456, 457] [428, 429, 458, 459] [430, 431, 460, 461] [432, 433, 462, 463] [434, 435, 464, 465] [436, 437, 466, 467] [438, 439, 468, 469] [440, 441, 470, 471] [442, 443, 472, 473] [444, 445, 474, 475] [446, 447, 476, 477] [448, 449, 478, 479] [451, 452, 481, 482] [453, 454, 483, 484] [455, 456, 485, 486] [457, 458, 487, 488] [459, 460, 489, 490] [461, 462, 491, 492] [463, 464, 493, 494] [465, 466, 495, 496] [467, 468, 497, 498] [469, 470, 499, 500] [471, 472, 501, 502] [473, 474, 503, 504] [475, 476, 505, 506] [477, 478, 507, 508] [450, 479, 480, 509] [480, 481, 510, 511] [482, 483, 512, 513] [484, 485, 514, 515] [486, 487, 516, 517] [488, 489, 518, 519] [490, 491, 520, 521] [492, 493, 522, 523] [494, 495, 524, 525] [496, 497, 526, 527] [498, 499, 528, 529] [500, 501, 530, 531] [502, 503, 532, 533] [504, 505, 534, 535] [506, 507, 536, 537] [508, 509, 538, 539] [511, 512, 541, 542] [513, 514, 543, 544] [515, 516, 545, 546] [517, 518, 547, 548] [519, 520, 549, 550] [521, 522, 551, 552] [523, 524, 553, 554] [525, 526, 555, 556] [527, 528, 557, 558] [529, 530, 559, 560] [531, 532, 561, 562] [533, 534, 563, 564] [535, 536, 565, 566] [537, 538, 567, 568] [510, 539, 540, 569] [540, 541, 570, 571] [542, 543, 572, 573] [544, 545, 574, 575] [546, 547, 576, 577] [548, 549, 578, 579] [550, 551, 580, 581] [552, 553, 582, 583] [554, 555, 584, 585] [556, 557, 586, 587] [558, 559, 588, 589] [560, 561, 590, 591] [562, 563, 592, 593] [564, 565, 594, 595] [566, 567, 596, 597] [568, 569, 598, 599] [571, 572, 601, 602] [573, 574, 603, 604] [575, 576, 605, 606] [577, 578, 607, 608] [579, 580, 609, 610] [581, 582, 611, 612] [583, 584, 613, 614] [585, 586, 615, 616] [587, 588, 617, 618] [589, 590, 619, 620] [591, 592, 621, 622] [593, 594, 623, 624] [595, 596, 625, 626] [597, 598, 627, 628] [570, 599, 600, 629] [600, 601, 630, 631] [602, 603, 632, 633] [604, 605, 634, 635] [606, 607, 636, 637] [608, 609, 638, 639] [610, 611, 640, 641] [612, 613, 642, 643] [614, 615, 644, 645] [616, 617, 646, 647] [618, 619, 648, 649] [620, 621, 650, 651] [622, 623, 652, 653] [624, 625, 654, 655] [626, 627, 656, 657] [628, 629, 658, 659] [631, 632, 661, 662] [633, 634, 663, 664] [635, 636, 665, 666] [637, 638, 667, 668] [639, 640, 669, 670] [641, 642, 671, 672] [643, 644, 673, 674] [645, 646, 675, 676] [647, 648, 677, 678] [649, 650, 679, 680] [651, 652, 681, 682] [653, 654, 683, 684] [655, 656, 685, 686] [657, 658, 687, 688] [630, 659, 660, 689] [660, 661, 690, 691] [662, 663, 692, 693] [664, 665, 694, 695] [666, 667, 696, 697] [668, 669, 698, 699] [670, 671, 700, 701] [672, 673, 702, 703] [674, 675, 704, 705] [676, 677, 706, 707] [678, 679, 708, 709] [680, 681, 710, 711] [682, 683, 712, 713] [684, 685, 714, 715] [686, 687, 716, 717] [688, 689, 718, 719] [691, 692, 721, 722] [693, 694, 723, 724] [695, 696, 725, 726] [697, 698, 727, 728] [699, 700, 729, 730] [701, 702, 731, 732] [703, 704, 733, 734] [705, 706, 735, 736] [707, 708, 737, 738] [709, 710, 739, 740] [711, 712, 741, 742] [713, 714, 743, 744] [715, 716, 745, 746] [717, 718, 747, 748] [690, 719, 720, 749] [720, 721, 750, 751] [722, 723, 752, 753] [724, 725, 754, 755] [726, 727, 756, 757] [728, 729, 758, 759] [730, 731, 760, 761] [732, 733, 762, 763] [734, 735, 764, 765] [736, 737, 766, 767] [738, 739, 768, 769] [740, 741, 770, 771] [742, 743, 772, 773] [744, 745, 774, 775] [746, 747, 776, 777] [748, 749, 778, 779] [751, 752, 781, 782] [753, 754, 783, 784] [755, 756, 785, 786] [757, 758, 787, 788] [759, 760, 789, 790] [761, 762, 791, 792] [763, 764, 793, 794] [765, 766, 795, 796] [767, 768, 797, 798] [769, 770, 799, 800] [771, 772, 801, 802] [773, 774, 803, 804] [775, 776, 805, 806] [777, 778, 807, 808] [750, 779, 780, 809] [780, 781, 810, 811] [782, 783, 812, 813] [784, 785, 814, 815] [786, 787, 816, 817] [788, 789, 818, 819] [790, 791, 820, 821] [792, 793, 822, 823] [794, 795, 824, 825] [796, 797, 826, 827] [798, 799, 828, 829] [800, 801, 830, 831] [802, 803, 832, 833] [804, 805, 834, 835] [806, 807, 836, 837] [808, 809, 838, 839] [811, 812, 841, 842] [813, 814, 843, 844] [815, 816, 845, 846] [817, 818, 847, 848] [819, 820, 849, 850] [821, 822, 851, 852] [823, 824, 853, 854] [825, 826, 855, 856] [827, 828, 857, 858] [829, 830, 859, 860] [831, 832, 861, 862] [833, 834, 863, 864] [835, 836, 865, 866] [837, 838, 867, 868] [810, 839, 840, 869] [840, 841, 870, 871] [842, 843, 872, 873] [844, 845, 874, 875] [846, 847, 876, 877] [848, 849, 878, 879] [850, 851, 880, 881] [852, 853, 882, 883] [854, 855, 884, 885] [856, 857, 886, 887] [858, 859, 888, 889] [860, 861, 890, 891] [862, 863, 892, 893] [864, 865, 894, 895] [866, 867, 896, 897] [868, 869, 898, 899] [1, 2, 871, 872] [3, 4, 873, 874] [5, 6, 875, 876] [7, 8, 877, 878] [9, 10, 879, 880] [11, 12, 881, 882] [13, 14, 883, 884] [15, 16, 885, 886] [17, 18, 887, 888] [19, 20, 889, 890] [21, 22, 891, 892] [23, 24, 893, 894] [25, 26, 895, 896] [27, 28, 897, 898] [0, 29, 870, 899]
H_Z (450 checks, sparse supports)
[1, 2, 31, 32] [3, 4, 33, 34] [5, 6, 35, 36] [7, 8, 37, 38] [9, 10, 39, 40] [11, 12, 41, 42] [13, 14, 43, 44] [15, 16, 45, 46] [17, 18, 47, 48] [19, 20, 49, 50] [21, 22, 51, 52] [23, 24, 53, 54] [25, 26, 55, 56] [27, 28, 57, 58] [0, 29, 30, 59] [30, 31, 60, 61] [32, 33, 62, 63] [34, 35, 64, 65] [36, 37, 66, 67] [38, 39, 68, 69] [40, 41, 70, 71] [42, 43, 72, 73] [44, 45, 74, 75] [46, 47, 76, 77] [48, 49, 78, 79] [50, 51, 80, 81] [52, 53, 82, 83] [54, 55, 84, 85] [56, 57, 86, 87] [58, 59, 88, 89] [61, 62, 91, 92] [63, 64, 93, 94] [65, 66, 95, 96] [67, 68, 97, 98] [69, 70, 99, 100] [71, 72, 101, 102] [73, 74, 103, 104] [75, 76, 105, 106] [77, 78, 107, 108] [79, 80, 109, 110] [81, 82, 111, 112] [83, 84, 113, 114] [85, 86, 115, 116] [87, 88, 117, 118] [60, 89, 90, 119] [90, 91, 120, 121] [92, 93, 122, 123] [94, 95, 124, 125] [96, 97, 126, 127] [98, 99, 128, 129] [100, 101, 130, 131] [102, 103, 132, 133] [104, 105, 134, 135] [106, 107, 136, 137] [108, 109, 138, 139] [110, 111, 140, 141] [112, 113, 142, 143] [114, 115, 144, 145] [116, 117, 146, 147] [118, 119, 148, 149] [121, 122, 151, 152] [123, 124, 153, 154] [125, 126, 155, 156] [127, 128, 157, 158] [129, 130, 159, 160] [131, 132, 161, 162] [133, 134, 163, 164] [135, 136, 165, 166] [137, 138, 167, 168] [139, 140, 169, 170] [141, 142, 171, 172] [143, 144, 173, 174] [145, 146, 175, 176] [147, 148, 177, 178] [120, 149, 150, 179] [150, 151, 180, 181] [152, 153, 182, 183] [154, 155, 184, 185] [156, 157, 186, 187] [158, 159, 188, 189] [160, 161, 190, 191] [162, 163, 192, 193] [164, 165, 194, 195] [166, 167, 196, 197] [168, 169, 198, 199] [170, 171, 200, 201] [172, 173, 202, 203] [174, 175, 204, 205] [176, 177, 206, 207] [178, 179, 208, 209] [181, 182, 211, 212] [183, 184, 213, 214] [185, 186, 215, 216] [187, 188, 217, 218] [189, 190, 219, 220] [191, 192, 221, 222] [193, 194, 223, 224] [195, 196, 225, 226] [197, 198, 227, 228] [199, 200, 229, 230] [201, 202, 231, 232] [203, 204, 233, 234] [205, 206, 235, 236] [207, 208, 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Code ID 900-2-30 · download JSON · raw on GitHub