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[[484,2,22]] d ≤
n
484
k
2
d
22
kd²/n
2.0
w
4
X/Z
1
g
0.125
r
2.8284
layers
1
swaps
1280

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Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · 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 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[2, 23, 44, 87, 108, 130, 152, 174, 196, 217, 238, 260, 283, 306, 329, 330, 353, 375, 397, 420, 443, 465]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[17, 39, 61, 83, 105, 126, 147, 168, 190, 212, 234, 257, 280, 303, 325, 346, 368, 391, 413, 434, 456, 479]
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)×484 (2,2)×1452 (3,2)×4356 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 484 (2,2): 1452 (3,2): 4356 (3,4): 968
trapping sets H_Z (1,2)×484 (2,2)×1452 (3,2)×4356 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 484 (2,2): 1452 (3,2): 4356 (3,4): 968
witness diameter X 21.2132 · Z 21.3776 (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 (484)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 1280 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 22x22 lattice: all plaquette checks, (i+j) even -> X; distance 22 (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

[[484,2,22]] 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 22 (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 <= 22. 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 = 22 (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 242 (max weight 4) · Z-checks 242 (max weight 4)
H_X (242 checks, sparse supports)
[0, 1, 22, 23] [2, 3, 24, 25] [4, 5, 26, 27] [6, 7, 28, 29] [8, 9, 30, 31] [10, 11, 32, 33] [12, 13, 34, 35] [14, 15, 36, 37] [16, 17, 38, 39] [18, 19, 40, 41] [20, 21, 42, 43] [23, 24, 45, 46] [25, 26, 47, 48] [27, 28, 49, 50] [29, 30, 51, 52] [31, 32, 53, 54] [33, 34, 55, 56] [35, 36, 57, 58] [37, 38, 59, 60] [39, 40, 61, 62] [41, 42, 63, 64] [22, 43, 44, 65] [44, 45, 66, 67] [46, 47, 68, 69] [48, 49, 70, 71] [50, 51, 72, 73] [52, 53, 74, 75] [54, 55, 76, 77] [56, 57, 78, 79] [58, 59, 80, 81] [60, 61, 82, 83] [62, 63, 84, 85] [64, 65, 86, 87] [67, 68, 89, 90] [69, 70, 91, 92] [71, 72, 93, 94] [73, 74, 95, 96] [75, 76, 97, 98] [77, 78, 99, 100] [79, 80, 101, 102] [81, 82, 103, 104] [83, 84, 105, 106] [85, 86, 107, 108] [66, 87, 88, 109] [88, 89, 110, 111] [90, 91, 112, 113] [92, 93, 114, 115] [94, 95, 116, 117] [96, 97, 118, 119] [98, 99, 120, 121] [100, 101, 122, 123] [102, 103, 124, 125] [104, 105, 126, 127] [106, 107, 128, 129] [108, 109, 130, 131] [111, 112, 133, 134] [113, 114, 135, 136] [115, 116, 137, 138] [117, 118, 139, 140] [119, 120, 141, 142] [121, 122, 143, 144] [123, 124, 145, 146] [125, 126, 147, 148] [127, 128, 149, 150] [129, 130, 151, 152] [110, 131, 132, 153] [132, 133, 154, 155] [134, 135, 156, 157] [136, 137, 158, 159] [138, 139, 160, 161] [140, 141, 162, 163] [142, 143, 164, 165] [144, 145, 166, 167] [146, 147, 168, 169] [148, 149, 170, 171] [150, 151, 172, 173] [152, 153, 174, 175] [155, 156, 177, 178] [157, 158, 179, 180] [159, 160, 181, 182] [161, 162, 183, 184] [163, 164, 185, 186] [165, 166, 187, 188] [167, 168, 189, 190] [169, 170, 191, 192] [171, 172, 193, 194] [173, 174, 195, 196] [154, 175, 176, 197] [176, 177, 198, 199] [178, 179, 200, 201] [180, 181, 202, 203] [182, 183, 204, 205] [184, 185, 206, 207] [186, 187, 208, 209] [188, 189, 210, 211] [190, 191, 212, 213] [192, 193, 214, 215] [194, 195, 216, 217] [196, 197, 218, 219] [199, 200, 221, 222] [201, 202, 223, 224] [203, 204, 225, 226] [205, 206, 227, 228] [207, 208, 229, 230] [209, 210, 231, 232] [211, 212, 233, 234] [213, 214, 235, 236] [215, 216, 237, 238] [217, 218, 239, 240] [198, 219, 220, 241] [220, 221, 242, 243] [222, 223, 244, 245] [224, 225, 246, 247] [226, 227, 248, 249] [228, 229, 250, 251] [230, 231, 252, 253] [232, 233, 254, 255] [234, 235, 256, 257] [236, 237, 258, 259] [238, 239, 260, 261] [240, 241, 262, 263] [243, 244, 265, 266] [245, 246, 267, 268] [247, 248, 269, 270] [249, 250, 271, 272] [251, 252, 273, 274] [253, 254, 275, 276] [255, 256, 277, 278] [257, 258, 279, 280] [259, 260, 281, 282] [261, 262, 283, 284] [242, 263, 264, 285] [264, 265, 286, 287] [266, 267, 288, 289] [268, 269, 290, 291] [270, 271, 292, 293] [272, 273, 294, 295] [274, 275, 296, 297] [276, 277, 298, 299] [278, 279, 300, 301] [280, 281, 302, 303] [282, 283, 304, 305] [284, 285, 306, 307] [287, 288, 309, 310] [289, 290, 311, 312] [291, 292, 313, 314] [293, 294, 315, 316] [295, 296, 317, 318] [297, 298, 319, 320] [299, 300, 321, 322] [301, 302, 323, 324] [303, 304, 325, 326] [305, 306, 327, 328] [286, 307, 308, 329] [308, 309, 330, 331] [310, 311, 332, 333] [312, 313, 334, 335] [314, 315, 336, 337] [316, 317, 338, 339] [318, 319, 340, 341] [320, 321, 342, 343] [322, 323, 344, 345] [324, 325, 346, 347] [326, 327, 348, 349] [328, 329, 350, 351] [331, 332, 353, 354] [333, 334, 355, 356] [335, 336, 357, 358] [337, 338, 359, 360] [339, 340, 361, 362] [341, 342, 363, 364] [343, 344, 365, 366] [345, 346, 367, 368] [347, 348, 369, 370] [349, 350, 371, 372] [330, 351, 352, 373] [352, 353, 374, 375] [354, 355, 376, 377] [356, 357, 378, 379] [358, 359, 380, 381] [360, 361, 382, 383] [362, 363, 384, 385] [364, 365, 386, 387] [366, 367, 388, 389] [368, 369, 390, 391] [370, 371, 392, 393] [372, 373, 394, 395] [375, 376, 397, 398] [377, 378, 399, 400] [379, 380, 401, 402] [381, 382, 403, 404] [383, 384, 405, 406] [385, 386, 407, 408] [387, 388, 409, 410] [389, 390, 411, 412] [391, 392, 413, 414] [393, 394, 415, 416] [374, 395, 396, 417] [396, 397, 418, 419] [398, 399, 420, 421] [400, 401, 422, 423] [402, 403, 424, 425] [404, 405, 426, 427] [406, 407, 428, 429] [408, 409, 430, 431] [410, 411, 432, 433] [412, 413, 434, 435] [414, 415, 436, 437] [416, 417, 438, 439] [419, 420, 441, 442] [421, 422, 443, 444] [423, 424, 445, 446] [425, 426, 447, 448] [427, 428, 449, 450] [429, 430, 451, 452] [431, 432, 453, 454] [433, 434, 455, 456] [435, 436, 457, 458] [437, 438, 459, 460] [418, 439, 440, 461] [440, 441, 462, 463] [442, 443, 464, 465] [444, 445, 466, 467] [446, 447, 468, 469] [448, 449, 470, 471] [450, 451, 472, 473] [452, 453, 474, 475] [454, 455, 476, 477] [456, 457, 478, 479] [458, 459, 480, 481] [460, 461, 482, 483] [1, 2, 463, 464] [3, 4, 465, 466] [5, 6, 467, 468] [7, 8, 469, 470] [9, 10, 471, 472] [11, 12, 473, 474] [13, 14, 475, 476] [15, 16, 477, 478] [17, 18, 479, 480] [19, 20, 481, 482] [0, 21, 462, 483]
H_Z (242 checks, sparse supports)
[1, 2, 23, 24] [3, 4, 25, 26] [5, 6, 27, 28] [7, 8, 29, 30] [9, 10, 31, 32] [11, 12, 33, 34] [13, 14, 35, 36] [15, 16, 37, 38] [17, 18, 39, 40] [19, 20, 41, 42] [0, 21, 22, 43] [22, 23, 44, 45] [24, 25, 46, 47] [26, 27, 48, 49] [28, 29, 50, 51] [30, 31, 52, 53] [32, 33, 54, 55] [34, 35, 56, 57] [36, 37, 58, 59] [38, 39, 60, 61] [40, 41, 62, 63] [42, 43, 64, 65] [45, 46, 67, 68] [47, 48, 69, 70] [49, 50, 71, 72] [51, 52, 73, 74] [53, 54, 75, 76] [55, 56, 77, 78] [57, 58, 79, 80] [59, 60, 81, 82] [61, 62, 83, 84] [63, 64, 85, 86] [44, 65, 66, 87] [66, 67, 88, 89] [68, 69, 90, 91] [70, 71, 92, 93] [72, 73, 94, 95] [74, 75, 96, 97] [76, 77, 98, 99] [78, 79, 100, 101] [80, 81, 102, 103] [82, 83, 104, 105] [84, 85, 106, 107] [86, 87, 108, 109] [89, 90, 111, 112] [91, 92, 113, 114] [93, 94, 115, 116] [95, 96, 117, 118] [97, 98, 119, 120] [99, 100, 121, 122] [101, 102, 123, 124] [103, 104, 125, 126] [105, 106, 127, 128] [107, 108, 129, 130] [88, 109, 110, 131] [110, 111, 132, 133] [112, 113, 134, 135] [114, 115, 136, 137] [116, 117, 138, 139] [118, 119, 140, 141] [120, 121, 142, 143] [122, 123, 144, 145] [124, 125, 146, 147] [126, 127, 148, 149] [128, 129, 150, 151] [130, 131, 152, 153] [133, 134, 155, 156] [135, 136, 157, 158] [137, 138, 159, 160] [139, 140, 161, 162] [141, 142, 163, 164] [143, 144, 165, 166] [145, 146, 167, 168] [147, 148, 169, 170] [149, 150, 171, 172] [151, 152, 173, 174] [132, 153, 154, 175] [154, 155, 176, 177] [156, 157, 178, 179] [158, 159, 180, 181] [160, 161, 182, 183] [162, 163, 184, 185] [164, 165, 186, 187] [166, 167, 188, 189] [168, 169, 190, 191] [170, 171, 192, 193] [172, 173, 194, 195] [174, 175, 196, 197] [177, 178, 199, 200] [179, 180, 201, 202] [181, 182, 203, 204] [183, 184, 205, 206] [185, 186, 207, 208] [187, 188, 209, 210] [189, 190, 211, 212] [191, 192, 213, 214] [193, 194, 215, 216] [195, 196, 217, 218] [176, 197, 198, 219] [198, 199, 220, 221] [200, 201, 222, 223] [202, 203, 224, 225] [204, 205, 226, 227] [206, 207, 228, 229] [208, 209, 230, 231] [210, 211, 232, 233] [212, 213, 234, 235] [214, 215, 236, 237] [216, 217, 238, 239] [218, 219, 240, 241] [221, 222, 243, 244] [223, 224, 245, 246] [225, 226, 247, 248] [227, 228, 249, 250] [229, 230, 251, 252] [231, 232, 253, 254] [233, 234, 255, 256] [235, 236, 257, 258] [237, 238, 259, 260] [239, 240, 261, 262] [220, 241, 242, 263] [242, 243, 264, 265] [244, 245, 266, 267] [246, 247, 268, 269] [248, 249, 270, 271] [250, 251, 272, 273] [252, 253, 274, 275] [254, 255, 276, 277] [256, 257, 278, 279] [258, 259, 280, 281] [260, 261, 282, 283] [262, 263, 284, 285] [265, 266, 287, 288] [267, 268, 289, 290] [269, 270, 291, 292] [271, 272, 293, 294] [273, 274, 295, 296] [275, 276, 297, 298] [277, 278, 299, 300] [279, 280, 301, 302] [281, 282, 303, 304] [283, 284, 305, 306] [264, 285, 286, 307] [286, 287, 308, 309] [288, 289, 310, 311] [290, 291, 312, 313] [292, 293, 314, 315] [294, 295, 316, 317] [296, 297, 318, 319] [298, 299, 320, 321] [300, 301, 322, 323] [302, 303, 324, 325] [304, 305, 326, 327] [306, 307, 328, 329] [309, 310, 331, 332] [311, 312, 333, 334] [313, 314, 335, 336] [315, 316, 337, 338] [317, 318, 339, 340] [319, 320, 341, 342] [321, 322, 343, 344] [323, 324, 345, 346] [325, 326, 347, 348] [327, 328, 349, 350] [308, 329, 330, 351] [330, 331, 352, 353] [332, 333, 354, 355] [334, 335, 356, 357] [336, 337, 358, 359] [338, 339, 360, 361] [340, 341, 362, 363] [342, 343, 364, 365] [344, 345, 366, 367] [346, 347, 368, 369] [348, 349, 370, 371] [350, 351, 372, 373] [353, 354, 375, 376] [355, 356, 377, 378] [357, 358, 379, 380] [359, 360, 381, 382] [361, 362, 383, 384] [363, 364, 385, 386] [365, 366, 387, 388] [367, 368, 389, 390] [369, 370, 391, 392] [371, 372, 393, 394] [352, 373, 374, 395] [374, 375, 396, 397] [376, 377, 398, 399] [378, 379, 400, 401] [380, 381, 402, 403] [382, 383, 404, 405] [384, 385, 406, 407] [386, 387, 408, 409] [388, 389, 410, 411] [390, 391, 412, 413] [392, 393, 414, 415] [394, 395, 416, 417] [397, 398, 419, 420] [399, 400, 421, 422] [401, 402, 423, 424] [403, 404, 425, 426] [405, 406, 427, 428] [407, 408, 429, 430] [409, 410, 431, 432] [411, 412, 433, 434] [413, 414, 435, 436] [415, 416, 437, 438] [396, 417, 418, 439] [418, 419, 440, 441] [420, 421, 442, 443] [422, 423, 444, 445] [424, 425, 446, 447] [426, 427, 448, 449] [428, 429, 450, 451] [430, 431, 452, 453] [432, 433, 454, 455] [434, 435, 456, 457] [436, 437, 458, 459] [438, 439, 460, 461] [441, 442, 463, 464] [443, 444, 465, 466] [445, 446, 467, 468] [447, 448, 469, 470] [449, 450, 471, 472] [451, 452, 473, 474] [453, 454, 475, 476] [455, 456, 477, 478] [457, 458, 479, 480] [459, 460, 481, 482] [440, 461, 462, 483] [0, 1, 462, 463] [2, 3, 464, 465] [4, 5, 466, 467] [6, 7, 468, 469] [8, 9, 470, 471] [10, 11, 472, 473] [12, 13, 474, 475] [14, 15, 476, 477] [16, 17, 478, 479] [18, 19, 480, 481] [20, 21, 482, 483]
Code ID 484-2-22 · download JSON · raw on GitHub