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[[485,5,11]] d =
n
485
k
5
d
11
kd²/n
1.247
w
4
X/Z
1
g
1.25
r
1.4142
layers
1
swaps
0

Share this result

Distance

X/Z asymmetry 1 · d_X = 11, d_Z = 11 · 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 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[7, 40, 73, 105, 138, 172, 207, 241, 275, 309, 344]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[189, 225, 260, 294, 329, 364, 392, 414, 435, 456, 478]
certificate exact, d = 11 · CryptoMiniSat 5.14.7 SAT
X: no logical < 11 exists; Z: no logical < 11 exists

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 3.758) · H_Z 2–4 (mean 3.758)
qubit degrees H_X 1–2 (mean 1.86) · H_Z 1–2 (mean 1.86)
trapping sets H_X (1,1)×68 (2,0)×28 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 68 (1,2): 417 (2,0): 28 (2,1): 136 (2,2): 1130 (3,0): 2 (3,1): 380 (3,2): 3109 (3,3): 74 (3,4): 712
trapping sets H_Z (1,1)×68 (2,0)×28 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 68 (1,2): 417 (2,0): 28 (2,1): 136 (2,2): 1130 (3,0): 2 (3,1): 380 (3,2): 3109 (3,3): 74 (3,4): 712
witness diameter X 10.4403 · Z 10.198 (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 (485)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 Dense-packed surface code of arXiv:2511.06758 (Fujiu et al.), distance 11: five rotated surface-code patches fused via code deformation into one weight-4 patch, reconstructed from the authors' released stim simulation. Matches the board's [[101,5,5]] and [[197,5,7]] members of the same family.
builds on https://arxiv.org/abs/2511.06758
date 2026-08-13
notes Extends the dense-packed surface-code family to distance 11. At r=sqrt(2), rho=1, g = kd^2/n = 1.247. Not dominated by any board code in the weight-4 x local-2d-single cell.
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

[[485,5,11]] — dense-packed surface code, distance 11

Direction & hypothesis

Companion to the d = 5, 7, 9 dense-packed submissions ([[101,5,5]], [[197,5,7]], [[325,5,9]]). Same construction family (arXiv:2511.06758, Fujiu et al., "Dense packing of the surface code") at distance 11. The paper evaluates dense packing at d = 5, 7, 9, 11; this is the d = 11 member: five logical qubits, weight-4 checks, single-layer honest layout, nearest-neighbour tilted lattice (r = √2).

What was searched

Reconstruction identical to notes/101-5-5.md, with distance = 11 in research/build_dense_surface.py (the five_dense_num mask scales with d). n = 485 data qubits, k = 5, weight 4.

GF(2) sanity: CSS commutation exact, rank(H_X) = rank(H_Z) = 240, so k = 485 − 240 − 240 = 5.

Evidence trail

  • CLI RIS gate (2000 trials) finds no logical lighter than 11 on either side:

    • d ≤ 11 (X and Z), witnesses of weight 11 on both sides.

  • The paper's own logical operators at d = 11 have length 11, so this is a

    • genuine [[485,5,11]] witness-backed upper bound.

  • Not exhaustively certified here (consistent with the d = 7, 9 members):

    • an exact d = 11 claim would need the repo's MILP certificate (verify/certify.py) or a deeper RIS refutation. Confidence: upper_bound.

Dead ends

Same k-counting / ancilla trap as the d = 5 instance (see notes/101-5-5.md); resolved by the verifier's data-qubit-only n rule. No density collapse at larger d: the shared-region structure persists and the n-savings vs standalone (5 × 121 = 605 data qubits) hold.

Tools

research/build_dense_surface.py (this repo) with argument 11; numpy GF(2) rref; ./qldpc submit (RIS witness search, schema, verifier, locality).

Reproduction

python research/build_dense_surface.py 11   # writes /tmp/dense_11.npz (hx, hz, coords)
./qldpc submit /tmp/dense_11.npz --coords /tmp/dense_11.npz --layers 1 \
  --authors @mathysrennela --family topological

Parity checks

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