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[[961,1,31]] d ≤
n
961
k
1
d
31
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 ≤ 31, d_Z ≤ 31 · 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 31 · witness weight 31 (claimed upper_bound)
witness operator (support, 31 qubits)
[403, 420, 421, 422, 423, 435, 436, 437, 438, 439, 450, 455, 460, 461, 462, 463, 471, 472, 473, 480, 487, 490, 495, 505, 510, 519, 520, 537, 540, 569, 570]
d_Z 31 · witness weight 31 (claimed upper_bound)
witness operator (support, 31 qubits)
[9, 39, 70, 101, 132, 163, 193, 223, 254, 285, 315, 346, 377, 408, 439, 469, 499, 529, 559, 590, 622, 653, 684, 715, 746, 777, 807, 838, 870, 901, 931]
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 3.875) · H_Z 2–4 (mean 3.875)
qubit degrees H_X 1–2 (mean 1.935) · H_Z 1–2 (mean 1.935)
trapping sets H_X (1,1)×62 (2,0)×30 (3,1)×358 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 62 (1,2): 899 (2,0): 30 (2,1): 122 (2,2): 2578 (3,1): 358 (3,2): 7445 (3,3): 60 (3,4): 1680
trapping sets H_Z (1,1)×62 (2,0)×30 (3,1)×358 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 62 (1,2): 899 (2,0): 30 (2,1): 122 (2,2): 2578 (3,1): 358 (3,2): 7445 (3,3): 60 (3,4): 1680
witness diameter X 30.0666 · Z 31.0483 (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 (961)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 31x31 grid: bulk 2x2 plaquette checks colored by (i+j) parity, weight-2 boundary pairs on all four sides; distance 31.
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

[[961,1,31]] 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 31 (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 <= 31. 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 = 31 (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 480 (max weight 4) · Z-checks 480 (max weight 4)
H_X (480 checks, sparse supports)
[0, 1, 31, 32] [2, 3, 33, 34] [4, 5, 35, 36] [6, 7, 37, 38] [8, 9, 39, 40] [10, 11, 41, 42] [12, 13, 43, 44] [14, 15, 45, 46] [16, 17, 47, 48] [18, 19, 49, 50] [20, 21, 51, 52] [22, 23, 53, 54] [24, 25, 55, 56] [26, 27, 57, 58] [28, 29, 59, 60] [32, 33, 63, 64] [34, 35, 65, 66] [36, 37, 67, 68] [38, 39, 69, 70] [40, 41, 71, 72] [42, 43, 73, 74] [44, 45, 75, 76] [46, 47, 77, 78] [48, 49, 79, 80] [50, 51, 81, 82] [52, 53, 83, 84] [54, 55, 85, 86] [56, 57, 87, 88] [58, 59, 89, 90] [60, 61, 91, 92] [62, 63, 93, 94] [64, 65, 95, 96] [66, 67, 97, 98] [68, 69, 99, 100] [70, 71, 101, 102] [72, 73, 103, 104] [74, 75, 105, 106] [76, 77, 107, 108] [78, 79, 109, 110] [80, 81, 111, 112] [82, 83, 113, 114] [84, 85, 115, 116] [86, 87, 117, 118] [88, 89, 119, 120] [90, 91, 121, 122] [94, 95, 125, 126] [96, 97, 127, 128] [98, 99, 129, 130] [100, 101, 131, 132] [102, 103, 133, 134] [104, 105, 135, 136] [106, 107, 137, 138] [108, 109, 139, 140] [110, 111, 141, 142] [112, 113, 143, 144] [114, 115, 145, 146] [116, 117, 147, 148] [118, 119, 149, 150] [120, 121, 151, 152] [122, 123, 153, 154] [124, 125, 155, 156] [126, 127, 157, 158] [128, 129, 159, 160] [130, 131, 161, 162] [132, 133, 163, 164] [134, 135, 165, 166] [136, 137, 167, 168] [138, 139, 169, 170] [140, 141, 171, 172] [142, 143, 173, 174] [144, 145, 175, 176] [146, 147, 177, 178] [148, 149, 179, 180] [150, 151, 181, 182] [152, 153, 183, 184] [156, 157, 187, 188] [158, 159, 189, 190] [160, 161, 191, 192] [162, 163, 193, 194] [164, 165, 195, 196] [166, 167, 197, 198] [168, 169, 199, 200] [170, 171, 201, 202] [172, 173, 203, 204] [174, 175, 205, 206] [176, 177, 207, 208] [178, 179, 209, 210] [180, 181, 211, 212] [182, 183, 213, 214] [184, 185, 215, 216] [186, 187, 217, 218] [188, 189, 219, 220] [190, 191, 221, 222] [192, 193, 223, 224] [194, 195, 225, 226] [196, 197, 227, 228] [198, 199, 229, 230] [200, 201, 231, 232] [202, 203, 233, 234] [204, 205, 235, 236] [206, 207, 237, 238] [208, 209, 239, 240] [210, 211, 241, 242] [212, 213, 243, 244] [214, 215, 245, 246] [218, 219, 249, 250] [220, 221, 251, 252] [222, 223, 253, 254] [224, 225, 255, 256] [226, 227, 257, 258] [228, 229, 259, 260] [230, 231, 261, 262] [232, 233, 263, 264] [234, 235, 265, 266] [236, 237, 267, 268] [238, 239, 269, 270] [240, 241, 271, 272] [242, 243, 273, 274] [244, 245, 275, 276] [246, 247, 277, 278] [248, 249, 279, 280] [250, 251, 281, 282] [252, 253, 283, 284] [254, 255, 285, 286] [256, 257, 287, 288] [258, 259, 289, 290] [260, 261, 291, 292] [262, 263, 293, 294] [264, 265, 295, 296] [266, 267, 297, 298] [268, 269, 299, 300] [270, 271, 301, 302] [272, 273, 303, 304] [274, 275, 305, 306] [276, 277, 307, 308] [280, 281, 311, 312] [282, 283, 313, 314] [284, 285, 315, 316] [286, 287, 317, 318] [288, 289, 319, 320] [290, 291, 321, 322] [292, 293, 323, 324] [294, 295, 325, 326] [296, 297, 327, 328] [298, 299, 329, 330] [300, 301, 331, 332] [302, 303, 333, 334] [304, 305, 335, 336] [306, 307, 337, 338] [308, 309, 339, 340] [310, 311, 341, 342] [312, 313, 343, 344] [314, 315, 345, 346] [316, 317, 347, 348] [318, 319, 349, 350] [320, 321, 351, 352] [322, 323, 353, 354] [324, 325, 355, 356] [326, 327, 357, 358] [328, 329, 359, 360] [330, 331, 361, 362] [332, 333, 363, 364] [334, 335, 365, 366] [336, 337, 367, 368] [338, 339, 369, 370] [342, 343, 373, 374] [344, 345, 375, 376] [346, 347, 377, 378] [348, 349, 379, 380] [350, 351, 381, 382] [352, 353, 383, 384] [354, 355, 385, 386] [356, 357, 387, 388] [358, 359, 389, 390] [360, 361, 391, 392] [362, 363, 393, 394] [364, 365, 395, 396] [366, 367, 397, 398] [368, 369, 399, 400] [370, 371, 401, 402] [372, 373, 403, 404] [374, 375, 405, 406] [376, 377, 407, 408] [378, 379, 409, 410] [380, 381, 411, 412] [382, 383, 413, 414] [384, 385, 415, 416] [386, 387, 417, 418] [388, 389, 419, 420] [390, 391, 421, 422] [392, 393, 423, 424] [394, 395, 425, 426] [396, 397, 427, 428] [398, 399, 429, 430] [400, 401, 431, 432] [404, 405, 435, 436] [406, 407, 437, 438] [408, 409, 439, 440] [410, 411, 441, 442] [412, 413, 443, 444] [414, 415, 445, 446] [416, 417, 447, 448] [418, 419, 449, 450] [420, 421, 451, 452] [422, 423, 453, 454] [424, 425, 455, 456] [426, 427, 457, 458] [428, 429, 459, 460] [430, 431, 461, 462] [432, 433, 463, 464] [434, 435, 465, 466] [436, 437, 467, 468] [438, 439, 469, 470] [440, 441, 471, 472] [442, 443, 473, 474] [444, 445, 475, 476] [446, 447, 477, 478] [448, 449, 479, 480] [450, 451, 481, 482] [452, 453, 483, 484] [454, 455, 485, 486] [456, 457, 487, 488] [458, 459, 489, 490] [460, 461, 491, 492] [462, 463, 493, 494] [466, 467, 497, 498] [468, 469, 499, 500] [470, 471, 501, 502] [472, 473, 503, 504] [474, 475, 505, 506] [476, 477, 507, 508] [478, 479, 509, 510] [480, 481, 511, 512] [482, 483, 513, 514] [484, 485, 515, 516] [486, 487, 517, 518] [488, 489, 519, 520] [490, 491, 521, 522] [492, 493, 523, 524] [494, 495, 525, 526] [496, 497, 527, 528] [498, 499, 529, 530] [500, 501, 531, 532] [502, 503, 533, 534] [504, 505, 535, 536] [506, 507, 537, 538] [508, 509, 539, 540] [510, 511, 541, 542] [512, 513, 543, 544] [514, 515, 545, 546] [516, 517, 547, 548] [518, 519, 549, 550] [520, 521, 551, 552] [522, 523, 553, 554] [524, 525, 555, 556] [528, 529, 559, 560] [530, 531, 561, 562] [532, 533, 563, 564] [534, 535, 565, 566] [536, 537, 567, 568] [538, 539, 569, 570] [540, 541, 571, 572] [542, 543, 573, 574] [544, 545, 575, 576] [546, 547, 577, 578] [548, 549, 579, 580] [550, 551, 581, 582] [552, 553, 583, 584] [554, 555, 585, 586] [556, 557, 587, 588] [558, 559, 589, 590] [560, 561, 591, 592] [562, 563, 593, 594] [564, 565, 595, 596] [566, 567, 597, 598] [568, 569, 599, 600] [570, 571, 601, 602] [572, 573, 603, 604] [574, 575, 605, 606] [576, 577, 607, 608] [578, 579, 609, 610] [580, 581, 611, 612] [582, 583, 613, 614] [584, 585, 615, 616] [586, 587, 617, 618] [590, 591, 621, 622] [592, 593, 623, 624] [594, 595, 625, 626] [596, 597, 627, 628] [598, 599, 629, 630] [600, 601, 631, 632] [602, 603, 633, 634] [604, 605, 635, 636] [606, 607, 637, 638] [608, 609, 639, 640] [610, 611, 641, 642] [612, 613, 643, 644] [614, 615, 645, 646] [616, 617, 647, 648] [618, 619, 649, 650] [620, 621, 651, 652] [622, 623, 653, 654] [624, 625, 655, 656] [626, 627, 657, 658] [628, 629, 659, 660] [630, 631, 661, 662] [632, 633, 663, 664] [634, 635, 665, 666] [636, 637, 667, 668] [638, 639, 669, 670] [640, 641, 671, 672] [642, 643, 673, 674] [644, 645, 675, 676] [646, 647, 677, 678] [648, 649, 679, 680] [652, 653, 683, 684] [654, 655, 685, 686] [656, 657, 687, 688] [658, 659, 689, 690] [660, 661, 691, 692] [662, 663, 693, 694] [664, 665, 695, 696] [666, 667, 697, 698] [668, 669, 699, 700] [670, 671, 701, 702] [672, 673, 703, 704] [674, 675, 705, 706] [676, 677, 707, 708] [678, 679, 709, 710] [680, 681, 711, 712] [682, 683, 713, 714] [684, 685, 715, 716] [686, 687, 717, 718] [688, 689, 719, 720] [690, 691, 721, 722] [692, 693, 723, 724] [694, 695, 725, 726] [696, 697, 727, 728] [698, 699, 729, 730] [700, 701, 731, 732] [702, 703, 733, 734] [704, 705, 735, 736] [706, 707, 737, 738] [708, 709, 739, 740] [710, 711, 741, 742] [714, 715, 745, 746] [716, 717, 747, 748] [718, 719, 749, 750] [720, 721, 751, 752] [722, 723, 753, 754] [724, 725, 755, 756] [726, 727, 757, 758] [728, 729, 759, 760] [730, 731, 761, 762] [732, 733, 763, 764] [734, 735, 765, 766] [736, 737, 767, 768] [738, 739, 769, 770] [740, 741, 771, 772] [742, 743, 773, 774] [744, 745, 775, 776] [746, 747, 777, 778] [748, 749, 779, 780] [750, 751, 781, 782] [752, 753, 783, 784] [754, 755, 785, 786] [756, 757, 787, 788] [758, 759, 789, 790] [760, 761, 791, 792] [762, 763, 793, 794] [764, 765, 795, 796] [766, 767, 797, 798] [768, 769, 799, 800] [770, 771, 801, 802] [772, 773, 803, 804] [776, 777, 807, 808] [778, 779, 809, 810] [780, 781, 811, 812] [782, 783, 813, 814] [784, 785, 815, 816] [786, 787, 817, 818] [788, 789, 819, 820] [790, 791, 821, 822] [792, 793, 823, 824] [794, 795, 825, 826] [796, 797, 827, 828] [798, 799, 829, 830] [800, 801, 831, 832] [802, 803, 833, 834] [804, 805, 835, 836] [806, 807, 837, 838] [808, 809, 839, 840] [810, 811, 841, 842] [812, 813, 843, 844] [814, 815, 845, 846] [816, 817, 847, 848] [818, 819, 849, 850] [820, 821, 851, 852] [822, 823, 853, 854] [824, 825, 855, 856] [826, 827, 857, 858] [828, 829, 859, 860] [830, 831, 861, 862] [832, 833, 863, 864] [834, 835, 865, 866] [838, 839, 869, 870] [840, 841, 871, 872] [842, 843, 873, 874] [844, 845, 875, 876] [846, 847, 877, 878] [848, 849, 879, 880] [850, 851, 881, 882] [852, 853, 883, 884] [854, 855, 885, 886] [856, 857, 887, 888] [858, 859, 889, 890] [860, 861, 891, 892] [862, 863, 893, 894] [864, 865, 895, 896] [866, 867, 897, 898] [868, 869, 899, 900] [870, 871, 901, 902] [872, 873, 903, 904] [874, 875, 905, 906] [876, 877, 907, 908] [878, 879, 909, 910] [880, 881, 911, 912] [882, 883, 913, 914] [884, 885, 915, 916] [886, 887, 917, 918] [888, 889, 919, 920] [890, 891, 921, 922] [892, 893, 923, 924] [894, 895, 925, 926] [896, 897, 927, 928] [900, 901, 931, 932] [902, 903, 933, 934] [904, 905, 935, 936] [906, 907, 937, 938] [908, 909, 939, 940] [910, 911, 941, 942] [912, 913, 943, 944] [914, 915, 945, 946] [916, 917, 947, 948] [918, 919, 949, 950] [920, 921, 951, 952] [922, 923, 953, 954] [924, 925, 955, 956] [926, 927, 957, 958] [928, 929, 959, 960] [30, 61] [31, 62] [92, 123] [93, 124] [154, 185] [155, 186] [216, 247] [217, 248] [278, 309] [279, 310] [340, 371] [341, 372] [402, 433] [403, 434] [464, 495] [465, 496] [526, 557] [527, 558] [588, 619] [589, 620] [650, 681] [651, 682] [712, 743] [713, 744] [774, 805] [775, 806] [836, 867] [837, 868] [898, 929] [899, 930]
H_Z (480 checks, sparse supports)
[1, 2, 32, 33] [3, 4, 34, 35] [5, 6, 36, 37] [7, 8, 38, 39] [9, 10, 40, 41] [11, 12, 42, 43] [13, 14, 44, 45] [15, 16, 46, 47] [17, 18, 48, 49] [19, 20, 50, 51] [21, 22, 52, 53] [23, 24, 54, 55] [25, 26, 56, 57] [27, 28, 58, 59] [29, 30, 60, 61] [31, 32, 62, 63] [33, 34, 64, 65] [35, 36, 66, 67] [37, 38, 68, 69] [39, 40, 70, 71] [41, 42, 72, 73] [43, 44, 74, 75] [45, 46, 76, 77] [47, 48, 78, 79] [49, 50, 80, 81] [51, 52, 82, 83] [53, 54, 84, 85] [55, 56, 86, 87] [57, 58, 88, 89] [59, 60, 90, 91] [63, 64, 94, 95] [65, 66, 96, 97] [67, 68, 98, 99] [69, 70, 100, 101] [71, 72, 102, 103] [73, 74, 104, 105] [75, 76, 106, 107] [77, 78, 108, 109] [79, 80, 110, 111] [81, 82, 112, 113] [83, 84, 114, 115] [85, 86, 116, 117] [87, 88, 118, 119] [89, 90, 120, 121] [91, 92, 122, 123] [93, 94, 124, 125] [95, 96, 126, 127] [97, 98, 128, 129] [99, 100, 130, 131] [101, 102, 132, 133] [103, 104, 134, 135] [105, 106, 136, 137] [107, 108, 138, 139] [109, 110, 140, 141] [111, 112, 142, 143] [113, 114, 144, 145] [115, 116, 146, 147] [117, 118, 148, 149] [119, 120, 150, 151] [121, 122, 152, 153] [125, 126, 156, 157] [127, 128, 158, 159] [129, 130, 160, 161] [131, 132, 162, 163] [133, 134, 164, 165] [135, 136, 166, 167] [137, 138, 168, 169] [139, 140, 170, 171] [141, 142, 172, 173] [143, 144, 174, 175] [145, 146, 176, 177] [147, 148, 178, 179] [149, 150, 180, 181] [151, 152, 182, 183] [153, 154, 184, 185] [155, 156, 186, 187] [157, 158, 188, 189] [159, 160, 190, 191] [161, 162, 192, 193] [163, 164, 194, 195] [165, 166, 196, 197] [167, 168, 198, 199] [169, 170, 200, 201] [171, 172, 202, 203] [173, 174, 204, 205] [175, 176, 206, 207] [177, 178, 208, 209] [179, 180, 210, 211] [181, 182, 212, 213] [183, 184, 214, 215] [187, 188, 218, 219] [189, 190, 220, 221] [191, 192, 222, 223] [193, 194, 224, 225] [195, 196, 226, 227] [197, 198, 228, 229] [199, 200, 230, 231] [201, 202, 232, 233] [203, 204, 234, 235] [205, 206, 236, 237] [207, 208, 238, 239] [209, 210, 240, 241] [211, 212, 242, 243] [213, 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Code ID 961-1-31 · download JSON · raw on GitHub