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[[441,1,21]] d ≤
n
441
k
1
d
21
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 ≤ 21, d_Z ≤ 21 · 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 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[32, 33, 52, 55, 56, 57, 72, 79, 92, 101, 108, 109, 110, 111, 112, 123, 128, 145, 146, 148, 168]
d_Z 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[10, 32, 53, 74, 96, 117, 137, 158, 179, 200, 222, 243, 263, 284, 305, 325, 346, 368, 389, 410, 432]
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 3.818) · H_Z 2–4 (mean 3.818)
qubit degrees H_X 1–2 (mean 1.905) · H_Z 1–2 (mean 1.905)
trapping sets H_X (1,1)×42 (2,0)×20 (3,1)×238 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 42 (1,2): 399 (2,0): 20 (2,1): 82 (2,2): 1118 (3,1): 238 (3,2): 3165 (3,3): 40 (3,4): 720
trapping sets H_Z (1,1)×42 (2,0)×20 (3,1)×238 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 42 (1,2): 399 (2,0): 20 (2,1): 82 (2,2): 1118 (3,1): 238 (3,2): 3165 (3,3): 40 (3,4): 720
witness diameter X 20.0998 · Z 20.0998 (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 (441)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 21x21 grid: bulk 2x2 plaquette checks colored by (i+j) parity, weight-2 boundary pairs on all four sides; distance 21.
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

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