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[[841,1,29]] d ≤
n
841
k
1
d
29
kd²/n
1.0
w
4
X/Z
1
g
1
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X ≤ 29, d_Z ≤ 29 · 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 29 · witness weight 29 (claimed upper_bound)
witness operator (support, 29 qubits)
[0, 1, 2, 3, 6, 7, 12, 13, 14, 15, 18, 19, 24, 25, 26, 27, 28, 33, 34, 37, 38, 39, 40, 45, 46, 49, 50, 51, 52]
d_Z 29 · witness weight 29 (claimed upper_bound)
witness operator (support, 29 qubits)
[4, 33, 62, 92, 122, 152, 181, 209, 238, 268, 297, 325, 353, 381, 409, 438, 467, 495, 524, 554, 583, 611, 639, 668, 698, 727, 755, 783, 812]
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 2–4 (mean 3.867) · H_Z 2–4 (mean 3.867)
qubit degrees H_X 1–2 (mean 1.931) · H_Z 1–2 (mean 1.931)
trapping sets H_X (1,1)×58 (2,0)×28 (3,1)×334 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 58 (1,2): 783 (2,0): 28 (2,1): 114 (2,2): 2238 (3,1): 334 (3,2): 6445 (3,3): 56 (3,4): 1456
trapping sets H_Z (1,1)×58 (2,0)×28 (3,1)×334 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 58 (1,2): 783 (2,0): 28 (2,1): 114 (2,2): 2238 (3,1): 334 (3,2): 6445 (3,3): 56 (3,4): 1456
witness diameter X 28.0 · Z 28.2843 (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 = 1.414
X checkZ checkqubit site (841)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 0 nearest-neighbor SWAPs per round in total, at most 0 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 surface code on an open 29x29 grid: bulk 2x2 plaquette checks colored by (i+j) parity, weight-2 boundary pairs on all four sides; distance 29.
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

[[841,1,29]] rotated surface 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 29 (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 <= 29. The distance is exact

    • by construction (no logical lighter than a minimum non-contractible chain exists on the open planar lattice), 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 surface code [[L^2,1,L]], L = 29 (odd):

  • qubits q = (i, j) -> q = i*L + j on an open L x L grid, unit-spaced layout

    • coordinates (i, j), single layer;

  • for each (i, j) in {0..L-2}^2 the 2x2 plaquette

    • {(i,j), (i,j+1), (i+1,j), (i+1,j+1)} (clipped to the grid) is an X check when i+j is even and a Z check when i+j is odd;

  • boundary pairs: plaquettes at (i, -1) for odd i and (i, L-1) for even i are

    • X checks; plaquettes at (-1, j) for even j and (L-1, j) for odd j are Z checks (each clips to a weight-2 pair);

  • (i+j) parity swapped gives the X<->Z-conjugated variant, an equivalent code.

    • Distance: a full row or column of plaquettes is a weight-L logical; none lighter exists (standard surface-code result).

Parity checks

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