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[[961,169,3]] d ≤
n
961
k
169
d
3
kd²/n
1.583
w
4
X/Z
1
g
1.58
r
1.4142
layers
1
swaps
10

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[73, 104, 135]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[896, 927, 957]
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 4 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.823) · H_Z 2–4 (mean 3.865)
qubit degrees H_X 1–3 (mean 1.615) · H_Z 1–2 (mean 1.553)
trapping sets H_X (1,1)×381 (2,0)×28 (3,0)×485 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 381 (1,2): 569 (1,3): 11 (2,0): 28 (2,1): 1064 (2,2): 1122 (2,3): 37 (3,0): 485 (3,1): 2112 (3,2): 2265 (3,3): 1077 (3,4): 417 (3,5): 11
trapping sets H_Z (1,1)×430 (2,0)×58 (3,0)×595 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 430 (1,2): 531 (2,0): 58 (2,1): 1126 (2,2): 1002 (3,0): 595 (3,1): 2128 (3,2): 1988 (3,3): 1022 (3,4): 326
witness diameter X 2.0 · Z 2.2361 (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 10 nearest-neighbor SWAPs per round in total, at most 1 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 Checkerboard plaquette code on 31x31 open square, mod-5 lattice phase sweep (px=2, pz=0), batched conflict resolution
date 2026-09-20
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

[[961,169,3]] — checkerboard plaquette code, mod-5 lattice phase sweep at W=31

Direction & hypothesis

Targeting the geometric-efficiency frontier (g = 9k/n for d=3, r=√2, ρ=1) in the local-2d-single × weight-4 track. The existing record was [[656,114,3]] (g = 1.564), a 26×26 checkerboard with two right-edge notches and 40 weight-2 boundary pins. The k-arithmetic of the scaled family is k = (2W−1) + H − W2 where H = holes and W2 = pins; the full-density mod-5 lattice has asymptote k/n → 1/5 (g → 1.8), so larger W should eventually beat the record. The question was whether W = 31 (n = 961, admissible under the extended tier: n ≤ 1000, w ≤ 8, d ≤ 40) was large enough.

What was searched

Full 25-phase sweep over (px, pz) ∈ {0,1,2,3,4}² at W = 31 (n = 961). Each phase: clean W×W square, X-face holes on mod-5 lattice (u + 2v ≡ px mod 5), Z-face holes on mirror (u − 2v ≡ pz mod 5), batched conflict resolution (weight-1 fixes → weight-2 pins, culprit-hole drops for unpartnerable exposures). Corner-notched variants (corner notch size 2–4) were also evaluated at the winning phase; all were worse than the clean square.

Evidence trail

Submitted code: phase (px=2, pz=0) at W=31.

  • Census: 0 weight-≤2 logicals on both sides (exact rowspace-aware audit).
  • RIS distance: dX = 3, dZ = 3 (200 trials per side).
  • k = 169, n = 961, 170 holes + 62 weight-2 pins.
  • g = 9 × 169 / 961 = 1.5827 > 1.564 (record beaten).

Full W=31 sweep results (top 5 clean phases):

| Phase | k | holes | pins | g | |-------|---|-------|------|-----| | (2,0) | 169 | 170 | 84 | 1.583 | | (0,4) | 163 | 169 | 93 | 1.527 | | (3,2) | 163 | 168 | 99 | 1.527 | | (4,1) | 163 | 168 | 100 | 1.527 | | (1,3) | 162 | 168 | 96 | 1.517 |

Corner-notched variants at W=31, phase (2,0):

  • corner=2: k=161, n=945, g=1.533 (clean)
  • corner=3: k=160, n=925, g=1.557 (not clean)
  • corner=4: k=149, n=897, g=1.495 (clean)

Gate verdict: passed: true. Weight class: weight-4. Locality class: local-2d-single. Board-advancing: yes. No exact duplicate or WL equivalent.

Dead ends

  • Default phase (3,2) at W=31 gives g = 1.527 — does not beat the record.

    • The phase choice is load-bearing; (2,0) is the unique winner.

  • W = 40 (n = 1600) sweep: best clean g = 1.547 — does not beat record

    • (the asymptote hasn't kicked in yet at these sizes; k/n is still below the 0.1738 threshold).

  • Corner notches reduce k more than they reduce n, so g drops.

Tools

GLM 5.3 Flash (Zed coding agent), research kit (geo656_scale2.py, geo656_phase_sweep.py, geo656_greedy.py), verify/validate_candidate.py. ~2 minutes total compute on Apple M-series.

Reproduction

# From the research kit:
from geo656_scale2 import build, resolve
from geo656_greedy import g_of
b = build(31, px=2, pz=0)
resolve(b)
# b.HX, b.HZ are the parity checks; b.k == 169, g_of(b.n, b.k, 3) == 1.58273

Parity checks

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