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[[529,1,23]] d ≤
n
529
k
1
d
23
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 ≤ 23, d_Z ≤ 23 · 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 23 · witness weight 23 (claimed upper_bound)
witness operator (support, 23 qubits)
[108, 109, 128, 129, 130, 133, 150, 157, 172, 181, 182, 183, 194, 216, 238, 260, 282, 299, 300, 301, 304, 325, 326]
d_Z 23 · witness weight 23 (claimed upper_bound)
witness operator (support, 23 qubits)
[3, 26, 49, 72, 96, 120, 143, 166, 189, 211, 234, 257, 280, 304, 328, 352, 376, 400, 423, 446, 469, 492, 516]
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 3.833) · H_Z 2–4 (mean 3.833)
qubit degrees H_X 1–2 (mean 1.913) · H_Z 1–2 (mean 1.913)
trapping sets H_X (1,1)×46 (2,0)×22 (3,1)×262 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 46 (1,2): 483 (2,0): 22 (2,1): 90 (2,2): 1362 (3,1): 262 (3,2): 3877 (3,3): 44 (3,4): 880
trapping sets H_Z (1,1)×46 (2,0)×22 (3,1)×262 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 46 (1,2): 483 (2,0): 22 (2,1): 90 (2,2): 1362 (3,1): 262 (3,2): 3877 (3,3): 44 (3,4): 880
witness diameter X 22.8035 · Z 23.0868 (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 (529)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 23x23 grid: bulk 2x2 plaquette checks colored by (i+j) parity, weight-2 boundary pairs on all four sides; distance 23.
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

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