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[[576,2,24]] d ≤
n
576
k
2
d
24
kd²/n
2.0
w
4
X/Z
1
g
0.125
r
2.8284
layers
1
swaps
1540

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Distance

X/Z asymmetry 1 · d_X ≤ 24, d_Z ≤ 24 · 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 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[18, 41, 65, 90, 115, 140, 165, 190, 214, 238, 262, 285, 309, 334, 358, 381, 405, 429, 452, 475, 499, 524, 548, 571]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[0, 25, 49, 73, 98, 123, 148, 172, 195, 218, 242, 267, 291, 315, 340, 364, 387, 410, 433, 456, 503, 526, 550, 575]
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)×576 (2,2)×1728 (3,2)×5184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 576 (2,2): 1728 (3,2): 5184 (3,4): 1152
trapping sets H_Z (1,2)×576 (2,2)×1728 (3,2)×5184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 576 (2,2): 1728 (3,2): 5184 (3,4): 1152
witness diameter X 23.7697 · Z 23.7697 (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 (576)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 1540 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 24x24 lattice: all plaquette checks, (i+j) even -> X; distance 24 (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

[[576,2,24]] 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 24 (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 <= 24. 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 = 24 (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 288 (max weight 4) · Z-checks 288 (max weight 4)
H_X (288 checks, sparse supports)
[0, 1, 24, 25] [2, 3, 26, 27] [4, 5, 28, 29] [6, 7, 30, 31] [8, 9, 32, 33] [10, 11, 34, 35] [12, 13, 36, 37] [14, 15, 38, 39] [16, 17, 40, 41] [18, 19, 42, 43] [20, 21, 44, 45] [22, 23, 46, 47] [25, 26, 49, 50] [27, 28, 51, 52] [29, 30, 53, 54] [31, 32, 55, 56] [33, 34, 57, 58] [35, 36, 59, 60] [37, 38, 61, 62] [39, 40, 63, 64] [41, 42, 65, 66] [43, 44, 67, 68] [45, 46, 69, 70] [24, 47, 48, 71] [48, 49, 72, 73] [50, 51, 74, 75] [52, 53, 76, 77] [54, 55, 78, 79] [56, 57, 80, 81] [58, 59, 82, 83] [60, 61, 84, 85] [62, 63, 86, 87] [64, 65, 88, 89] [66, 67, 90, 91] [68, 69, 92, 93] [70, 71, 94, 95] [73, 74, 97, 98] [75, 76, 99, 100] [77, 78, 101, 102] [79, 80, 103, 104] [81, 82, 105, 106] [83, 84, 107, 108] [85, 86, 109, 110] [87, 88, 111, 112] [89, 90, 113, 114] [91, 92, 115, 116] [93, 94, 117, 118] [72, 95, 96, 119] [96, 97, 120, 121] [98, 99, 122, 123] [100, 101, 124, 125] [102, 103, 126, 127] [104, 105, 128, 129] [106, 107, 130, 131] [108, 109, 132, 133] [110, 111, 134, 135] [112, 113, 136, 137] [114, 115, 138, 139] [116, 117, 140, 141] [118, 119, 142, 143] [121, 122, 145, 146] [123, 124, 147, 148] [125, 126, 149, 150] [127, 128, 151, 152] [129, 130, 153, 154] [131, 132, 155, 156] [133, 134, 157, 158] [135, 136, 159, 160] [137, 138, 161, 162] [139, 140, 163, 164] [141, 142, 165, 166] [120, 143, 144, 167] [144, 145, 168, 169] [146, 147, 170, 171] [148, 149, 172, 173] [150, 151, 174, 175] [152, 153, 176, 177] [154, 155, 178, 179] [156, 157, 180, 181] [158, 159, 182, 183] [160, 161, 184, 185] [162, 163, 186, 187] [164, 165, 188, 189] [166, 167, 190, 191] [169, 170, 193, 194] [171, 172, 195, 196] [173, 174, 197, 198] [175, 176, 199, 200] [177, 178, 201, 202] [179, 180, 203, 204] [181, 182, 205, 206] [183, 184, 207, 208] [185, 186, 209, 210] [187, 188, 211, 212] [189, 190, 213, 214] [168, 191, 192, 215] [192, 193, 216, 217] [194, 195, 218, 219] [196, 197, 220, 221] [198, 199, 222, 223] [200, 201, 224, 225] [202, 203, 226, 227] [204, 205, 228, 229] [206, 207, 230, 231] [208, 209, 232, 233] [210, 211, 234, 235] [212, 213, 236, 237] [214, 215, 238, 239] [217, 218, 241, 242] [219, 220, 243, 244] [221, 222, 245, 246] [223, 224, 247, 248] [225, 226, 249, 250] [227, 228, 251, 252] [229, 230, 253, 254] [231, 232, 255, 256] [233, 234, 257, 258] [235, 236, 259, 260] [237, 238, 261, 262] [216, 239, 240, 263] [240, 241, 264, 265] [242, 243, 266, 267] [244, 245, 268, 269] [246, 247, 270, 271] [248, 249, 272, 273] [250, 251, 274, 275] [252, 253, 276, 277] [254, 255, 278, 279] [256, 257, 280, 281] [258, 259, 282, 283] [260, 261, 284, 285] [262, 263, 286, 287] [265, 266, 289, 290] [267, 268, 291, 292] [269, 270, 293, 294] [271, 272, 295, 296] [273, 274, 297, 298] [275, 276, 299, 300] [277, 278, 301, 302] [279, 280, 303, 304] [281, 282, 305, 306] [283, 284, 307, 308] [285, 286, 309, 310] [264, 287, 288, 311] [288, 289, 312, 313] [290, 291, 314, 315] [292, 293, 316, 317] [294, 295, 318, 319] [296, 297, 320, 321] [298, 299, 322, 323] [300, 301, 324, 325] [302, 303, 326, 327] [304, 305, 328, 329] [306, 307, 330, 331] [308, 309, 332, 333] [310, 311, 334, 335] [313, 314, 337, 338] [315, 316, 339, 340] [317, 318, 341, 342] [319, 320, 343, 344] [321, 322, 345, 346] [323, 324, 347, 348] [325, 326, 349, 350] [327, 328, 351, 352] [329, 330, 353, 354] [331, 332, 355, 356] [333, 334, 357, 358] [312, 335, 336, 359] [336, 337, 360, 361] [338, 339, 362, 363] [340, 341, 364, 365] [342, 343, 366, 367] [344, 345, 368, 369] [346, 347, 370, 371] [348, 349, 372, 373] [350, 351, 374, 375] [352, 353, 376, 377] [354, 355, 378, 379] [356, 357, 380, 381] [358, 359, 382, 383] [361, 362, 385, 386] [363, 364, 387, 388] [365, 366, 389, 390] [367, 368, 391, 392] [369, 370, 393, 394] [371, 372, 395, 396] [373, 374, 397, 398] [375, 376, 399, 400] [377, 378, 401, 402] [379, 380, 403, 404] [381, 382, 405, 406] [360, 383, 384, 407] [384, 385, 408, 409] [386, 387, 410, 411] [388, 389, 412, 413] [390, 391, 414, 415] [392, 393, 416, 417] [394, 395, 418, 419] [396, 397, 420, 421] [398, 399, 422, 423] [400, 401, 424, 425] [402, 403, 426, 427] [404, 405, 428, 429] [406, 407, 430, 431] [409, 410, 433, 434] [411, 412, 435, 436] [413, 414, 437, 438] [415, 416, 439, 440] [417, 418, 441, 442] [419, 420, 443, 444] [421, 422, 445, 446] [423, 424, 447, 448] [425, 426, 449, 450] [427, 428, 451, 452] [429, 430, 453, 454] [408, 431, 432, 455] [432, 433, 456, 457] [434, 435, 458, 459] [436, 437, 460, 461] [438, 439, 462, 463] [440, 441, 464, 465] [442, 443, 466, 467] [444, 445, 468, 469] [446, 447, 470, 471] [448, 449, 472, 473] [450, 451, 474, 475] [452, 453, 476, 477] [454, 455, 478, 479] [457, 458, 481, 482] [459, 460, 483, 484] [461, 462, 485, 486] [463, 464, 487, 488] [465, 466, 489, 490] [467, 468, 491, 492] [469, 470, 493, 494] [471, 472, 495, 496] [473, 474, 497, 498] [475, 476, 499, 500] [477, 478, 501, 502] [456, 479, 480, 503] [480, 481, 504, 505] [482, 483, 506, 507] [484, 485, 508, 509] [486, 487, 510, 511] [488, 489, 512, 513] [490, 491, 514, 515] [492, 493, 516, 517] [494, 495, 518, 519] [496, 497, 520, 521] [498, 499, 522, 523] [500, 501, 524, 525] [502, 503, 526, 527] [505, 506, 529, 530] [507, 508, 531, 532] [509, 510, 533, 534] [511, 512, 535, 536] [513, 514, 537, 538] [515, 516, 539, 540] [517, 518, 541, 542] [519, 520, 543, 544] [521, 522, 545, 546] [523, 524, 547, 548] [525, 526, 549, 550] [504, 527, 528, 551] [528, 529, 552, 553] [530, 531, 554, 555] [532, 533, 556, 557] [534, 535, 558, 559] [536, 537, 560, 561] [538, 539, 562, 563] [540, 541, 564, 565] [542, 543, 566, 567] [544, 545, 568, 569] [546, 547, 570, 571] [548, 549, 572, 573] [550, 551, 574, 575] [1, 2, 553, 554] [3, 4, 555, 556] [5, 6, 557, 558] [7, 8, 559, 560] [9, 10, 561, 562] [11, 12, 563, 564] [13, 14, 565, 566] [15, 16, 567, 568] [17, 18, 569, 570] [19, 20, 571, 572] [21, 22, 573, 574] [0, 23, 552, 575]
H_Z (288 checks, sparse supports)
[1, 2, 25, 26] [3, 4, 27, 28] [5, 6, 29, 30] [7, 8, 31, 32] [9, 10, 33, 34] [11, 12, 35, 36] [13, 14, 37, 38] [15, 16, 39, 40] [17, 18, 41, 42] [19, 20, 43, 44] [21, 22, 45, 46] [0, 23, 24, 47] [24, 25, 48, 49] [26, 27, 50, 51] [28, 29, 52, 53] [30, 31, 54, 55] [32, 33, 56, 57] [34, 35, 58, 59] [36, 37, 60, 61] [38, 39, 62, 63] [40, 41, 64, 65] [42, 43, 66, 67] [44, 45, 68, 69] [46, 47, 70, 71] [49, 50, 73, 74] [51, 52, 75, 76] [53, 54, 77, 78] [55, 56, 79, 80] [57, 58, 81, 82] [59, 60, 83, 84] [61, 62, 85, 86] [63, 64, 87, 88] [65, 66, 89, 90] [67, 68, 91, 92] [69, 70, 93, 94] [48, 71, 72, 95] [72, 73, 96, 97] [74, 75, 98, 99] [76, 77, 100, 101] [78, 79, 102, 103] [80, 81, 104, 105] [82, 83, 106, 107] [84, 85, 108, 109] [86, 87, 110, 111] [88, 89, 112, 113] [90, 91, 114, 115] [92, 93, 116, 117] [94, 95, 118, 119] [97, 98, 121, 122] [99, 100, 123, 124] [101, 102, 125, 126] [103, 104, 127, 128] [105, 106, 129, 130] [107, 108, 131, 132] [109, 110, 133, 134] [111, 112, 135, 136] [113, 114, 137, 138] [115, 116, 139, 140] [117, 118, 141, 142] [96, 119, 120, 143] [120, 121, 144, 145] [122, 123, 146, 147] [124, 125, 148, 149] [126, 127, 150, 151] [128, 129, 152, 153] [130, 131, 154, 155] [132, 133, 156, 157] [134, 135, 158, 159] [136, 137, 160, 161] [138, 139, 162, 163] [140, 141, 164, 165] [142, 143, 166, 167] [145, 146, 169, 170] [147, 148, 171, 172] [149, 150, 173, 174] [151, 152, 175, 176] [153, 154, 177, 178] [155, 156, 179, 180] [157, 158, 181, 182] [159, 160, 183, 184] [161, 162, 185, 186] [163, 164, 187, 188] [165, 166, 189, 190] [144, 167, 168, 191] [168, 169, 192, 193] [170, 171, 194, 195] [172, 173, 196, 197] [174, 175, 198, 199] [176, 177, 200, 201] [178, 179, 202, 203] [180, 181, 204, 205] [182, 183, 206, 207] [184, 185, 208, 209] [186, 187, 210, 211] [188, 189, 212, 213] [190, 191, 214, 215] [193, 194, 217, 218] [195, 196, 219, 220] [197, 198, 221, 222] [199, 200, 223, 224] [201, 202, 225, 226] [203, 204, 227, 228] [205, 206, 229, 230] [207, 208, 231, 232] [209, 210, 233, 234] [211, 212, 235, 236] [213, 214, 237, 238] [192, 215, 216, 239] [216, 217, 240, 241] [218, 219, 242, 243] [220, 221, 244, 245] [222, 223, 246, 247] [224, 225, 248, 249] [226, 227, 250, 251] [228, 229, 252, 253] [230, 231, 254, 255] [232, 233, 256, 257] [234, 235, 258, 259] [236, 237, 260, 261] [238, 239, 262, 263] [241, 242, 265, 266] [243, 244, 267, 268] [245, 246, 269, 270] [247, 248, 271, 272] [249, 250, 273, 274] [251, 252, 275, 276] [253, 254, 277, 278] [255, 256, 279, 280] [257, 258, 281, 282] [259, 260, 283, 284] [261, 262, 285, 286] [240, 263, 264, 287] [264, 265, 288, 289] [266, 267, 290, 291] [268, 269, 292, 293] [270, 271, 294, 295] [272, 273, 296, 297] [274, 275, 298, 299] [276, 277, 300, 301] [278, 279, 302, 303] [280, 281, 304, 305] [282, 283, 306, 307] [284, 285, 308, 309] [286, 287, 310, 311] [289, 290, 313, 314] [291, 292, 315, 316] [293, 294, 317, 318] [295, 296, 319, 320] [297, 298, 321, 322] [299, 300, 323, 324] [301, 302, 325, 326] [303, 304, 327, 328] [305, 306, 329, 330] [307, 308, 331, 332] [309, 310, 333, 334] [288, 311, 312, 335] [312, 313, 336, 337] [314, 315, 338, 339] [316, 317, 340, 341] [318, 319, 342, 343] [320, 321, 344, 345] [322, 323, 346, 347] [324, 325, 348, 349] [326, 327, 350, 351] [328, 329, 352, 353] [330, 331, 354, 355] [332, 333, 356, 357] [334, 335, 358, 359] [337, 338, 361, 362] [339, 340, 363, 364] [341, 342, 365, 366] [343, 344, 367, 368] [345, 346, 369, 370] [347, 348, 371, 372] [349, 350, 373, 374] [351, 352, 375, 376] [353, 354, 377, 378] [355, 356, 379, 380] [357, 358, 381, 382] [336, 359, 360, 383] [360, 361, 384, 385] [362, 363, 386, 387] [364, 365, 388, 389] [366, 367, 390, 391] [368, 369, 392, 393] [370, 371, 394, 395] [372, 373, 396, 397] [374, 375, 398, 399] [376, 377, 400, 401] [378, 379, 402, 403] [380, 381, 404, 405] [382, 383, 406, 407] [385, 386, 409, 410] [387, 388, 411, 412] [389, 390, 413, 414] [391, 392, 415, 416] [393, 394, 417, 418] [395, 396, 419, 420] [397, 398, 421, 422] [399, 400, 423, 424] [401, 402, 425, 426] [403, 404, 427, 428] [405, 406, 429, 430] [384, 407, 408, 431] [408, 409, 432, 433] [410, 411, 434, 435] [412, 413, 436, 437] [414, 415, 438, 439] [416, 417, 440, 441] [418, 419, 442, 443] [420, 421, 444, 445] [422, 423, 446, 447] [424, 425, 448, 449] [426, 427, 450, 451] [428, 429, 452, 453] [430, 431, 454, 455] [433, 434, 457, 458] [435, 436, 459, 460] [437, 438, 461, 462] [439, 440, 463, 464] [441, 442, 465, 466] [443, 444, 467, 468] [445, 446, 469, 470] [447, 448, 471, 472] [449, 450, 473, 474] [451, 452, 475, 476] [453, 454, 477, 478] [432, 455, 456, 479] [456, 457, 480, 481] [458, 459, 482, 483] [460, 461, 484, 485] [462, 463, 486, 487] [464, 465, 488, 489] [466, 467, 490, 491] [468, 469, 492, 493] [470, 471, 494, 495] [472, 473, 496, 497] [474, 475, 498, 499] [476, 477, 500, 501] [478, 479, 502, 503] [481, 482, 505, 506] [483, 484, 507, 508] [485, 486, 509, 510] [487, 488, 511, 512] [489, 490, 513, 514] [491, 492, 515, 516] [493, 494, 517, 518] [495, 496, 519, 520] [497, 498, 521, 522] [499, 500, 523, 524] [501, 502, 525, 526] [480, 503, 504, 527] [504, 505, 528, 529] [506, 507, 530, 531] [508, 509, 532, 533] [510, 511, 534, 535] [512, 513, 536, 537] [514, 515, 538, 539] [516, 517, 540, 541] [518, 519, 542, 543] [520, 521, 544, 545] [522, 523, 546, 547] [524, 525, 548, 549] [526, 527, 550, 551] [529, 530, 553, 554] [531, 532, 555, 556] [533, 534, 557, 558] [535, 536, 559, 560] [537, 538, 561, 562] [539, 540, 563, 564] [541, 542, 565, 566] [543, 544, 567, 568] [545, 546, 569, 570] [547, 548, 571, 572] [549, 550, 573, 574] [528, 551, 552, 575] [0, 1, 552, 553] [2, 3, 554, 555] [4, 5, 556, 557] [6, 7, 558, 559] [8, 9, 560, 561] [10, 11, 562, 563] [12, 13, 564, 565] [14, 15, 566, 567] [16, 17, 568, 569] [18, 19, 570, 571] [20, 21, 572, 573] [22, 23, 574, 575]
Code ID 576-2-24 · download JSON · raw on GitHub