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[[677,5,13]] d =
n
677
k
5
d
13
kd²/n
1.248
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 = 13, d_Z = 13 · 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 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[9, 48, 87, 126, 166, 206, 245, 285, 325, 366, 407, 447, 487]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[246, 287, 328, 368, 409, 451, 492, 526, 553, 580, 606, 631, 657]
certificate exact, d = 13 · CryptoMiniSat 5.14.7 SAT
X: no logical < 13 exists; Z: no logical < 13 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 3.792) · H_Z 2–4 (mean 3.792)
qubit degrees H_X 1–2 (mean 1.882) · H_Z 1–2 (mean 1.882)
trapping sets H_X (1,1)×80 (2,0)×34 (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): 80 (1,2): 597 (2,0): 34 (2,1): 160 (2,2): 1646 (3,0): 2 (3,1): 452 (3,2): 4597 (3,3): 86 (3,4): 1048
trapping sets H_Z (1,1)×80 (2,0)×34 (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): 80 (1,2): 597 (2,0): 34 (2,1): 160 (2,2): 1646 (3,0): 2 (3,1): 452 (3,2): 4597 (3,3): 86 (3,4): 1048
witness diameter X 12.1655 · Z 12.0416 (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 (677)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 13: 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 13, the largest member under the n<=700 cap. At r=sqrt(2), rho=1, g = kd^2/n = 1.248. 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

[[677,5,13]] — dense-packed surface code, distance 13

Direction & hypothesis

Companion to the d = 5, 7, 9, 11 dense-packed submissions ([[101,5,5]], [[197,5,7]], [[325,5,9]], [[485,5,11]]). Same construction family (arXiv:2511.06758, Fujiu et al., "Dense packing of the surface code") at distance 13. Five logical qubits, weight-4 checks, single-layer honest layout, nearest-neighbour tilted lattice (r = √2). This is the largest member that fits under the n ≤ 700 cap (n = 677).

What was searched

Reconstruction identical to notes/101-5-5.md, with distance = 13 in research/build_dense_surface.py (the five_dense_num mask scales with d). n = 677 data qubits, k = 5, weight 4.

GF(2) sanity: CSS commutation exact, rank(H_X) = rank(H_Z) = 336, so k = 677 − 336 − 336 = 5.

Evidence trail

  • CLI RIS gate (2000 trials) finds no logical lighter than 13 on either side:

    • d ≤ 13 (X and Z), witnesses of weight 13 on both sides.

  • The paper's own logical operators at d = 13 have length 13, so this is a

    • genuine [[677,5,13]] witness-backed upper bound.

  • Not exhaustively certified here (consistent with the d = 7, 9, 11

    • members): an exact d = 13 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 × 169 = 845 data qubits) hold.

Tools

research/build_dense_surface.py (this repo) with argument 13; numpy GF(2) rref; ./qldpc submit (RIS witness search, schema, verifier, locality).

Reproduction

python research/build_dense_surface.py 13   # writes /tmp/dense_13.npz (hx, hz, coords)
./qldpc submit /tmp/dense_13.npz --coords /tmp/dense_13.npz --layers 1 \
  --authors @mathysrennela --family topological

Parity checks

X-checks 336 (max weight 4) · Z-checks 336 (max weight 4)
H_X (336 checks, sparse supports)
[0, 1] [2, 3] [4, 5] [6, 7] [8, 9] [10, 11] [13, 14] [15, 16] [17, 18] [19, 20] [21, 22] [23, 24] [26, 27] [28, 29] [30, 31] [32, 33] [34, 35] [36, 37] [1, 2, 40, 41] [3, 4, 42, 43] [5, 6, 44, 45] [7, 8, 46, 47] [9, 10, 48, 49] [11, 12, 50, 51] [14, 15, 53, 54] [16, 17, 55, 56] [18, 19, 57, 58] [20, 21, 59, 60] [22, 23, 61, 62] [24, 25, 63, 64] [27, 28, 66, 67] [29, 30, 68, 69] [31, 32, 70, 71] [33, 34, 72, 73] [35, 36, 74, 75] [37, 38, 76, 77] [39, 40, 78, 79] [41, 42, 80, 81] [43, 44, 82, 83] [45, 46, 84, 85] [47, 48, 86, 87] [49, 50, 88, 89] [52, 53, 91, 92] [54, 55, 93, 94] [56, 57, 95, 96] [58, 59, 97, 98] [60, 61, 99, 100] [62, 63, 101, 102] [65, 66, 104, 105] [67, 68, 106, 107] [69, 70, 108, 109] [71, 72, 110, 111] [73, 74, 112, 113] [75, 76, 114, 115] [79, 80, 118, 119] [81, 82, 120, 121] [83, 84, 122, 123] [85, 86, 124, 125] [87, 88, 126, 127] [89, 90, 128, 129] [92, 93, 131, 132] [94, 95, 133, 134] [96, 97, 135, 136] [98, 99, 137, 138] [100, 101, 139, 140] [102, 103, 141, 142] [105, 106, 144, 145] [107, 108, 146, 147] [109, 110, 148, 149] [111, 112, 150, 151] [113, 114, 152, 153] [115, 116, 154, 155] [117, 118, 156, 157] [119, 120, 158, 159] [121, 122, 160, 161] [123, 124, 162, 163] [125, 126, 164, 165] [127, 128, 166, 167] [130, 131, 169, 170] [132, 133, 171, 172] [134, 135, 173, 174] [136, 137, 175, 176] [138, 139, 177, 178] [140, 141, 179, 180] [143, 144, 182, 183] [145, 146, 184, 185] [147, 148, 186, 187] [149, 150, 188, 189] [151, 152, 190, 191] [153, 154, 192, 193] [157, 158, 196, 197] [159, 160, 198, 199] [161, 162, 200, 201] [163, 164, 202, 203] [165, 166, 204, 205] [167, 168, 206, 207] [170, 171, 209, 210] [172, 173, 211, 212] [174, 175, 213, 214] [176, 177, 215, 216] [178, 179, 217, 218] [180, 181, 219, 220] [183, 184, 222, 223] [185, 186, 224, 225] [187, 188, 226, 227] [189, 190, 228, 229] [191, 192, 230, 231] [193, 194, 232, 233] [195, 196, 234, 235] [197, 198, 236, 237] [199, 200, 238, 239] [201, 202, 240, 241] [203, 204, 242, 243] [205, 206, 244, 245] [208, 209, 248, 249] [210, 211, 250, 251] [212, 213, 252, 253] [214, 215, 254, 255] [216, 217, 256, 257] [218, 219, 258, 259] [221, 222, 262, 263] [223, 224, 264, 265] [225, 226, 266, 267] [227, 228, 268, 269] [229, 230, 270, 271] [231, 232, 272, 273] [235, 236, 276, 277] [237, 238, 278, 279] [239, 240, 280, 281] [241, 242, 282, 283] [243, 244, 284, 285] [245, 246, 286, 287] [247, 248, 288, 289] [249, 250, 290, 291] [251, 252, 292, 293] [253, 254, 294, 295] [255, 256, 296, 297] [257, 258, 298, 299] [259, 260, 300, 301] [261, 262, 302, 303] [263, 264, 304, 305] [265, 266, 306, 307] [267, 268, 308, 309] [269, 270, 310, 311] [271, 272, 312, 313] [273, 274, 314, 315] [275, 276, 316, 317] [277, 278, 318, 319] [279, 280, 320, 321] [281, 282, 322, 323] [283, 284, 324, 325] [285, 286, 326, 327] [287, 288, 328, 329] [289, 290, 330, 331] [291, 292, 332, 333] [293, 294, 334, 335] [295, 296, 336, 337] [297, 298, 338, 339] [299, 300, 340, 341] [301, 302, 342, 343] [303, 304, 344, 345] [305, 306, 346, 347] [307, 308, 348, 349] [309, 310, 350, 351] [311, 312, 352, 353] [313, 314, 354, 355] [317, 318, 358, 359] [319, 320, 360, 361] [321, 322, 362, 363] [323, 324, 364, 365] [325, 326, 366, 367] [327, 328, 368, 369] [329, 330, 370, 371] [331, 332, 372, 373] [333, 334, 374, 375] [335, 336, 376, 377] [337, 338, 378, 379] [339, 340, 380, 381] [341, 342, 382, 383] [343, 344, 384, 385] [345, 346, 386, 387] [347, 348, 388, 389] [349, 350, 390, 391] [351, 352, 392, 393] [353, 354, 394, 395] [355, 356, 396, 397] [357, 358, 398, 399] [359, 360, 400, 401] [361, 362, 402, 403] [363, 364, 404, 405] [365, 366, 406, 407] [367, 368, 408, 409] [369, 370, 410, 411] [371, 372, 412, 413] [373, 374, 414, 415] [375, 376, 416, 417] [377, 378, 418, 419] [379, 380, 420, 421] [381, 382, 422, 423] [383, 384, 424, 425] [385, 386, 426, 427] [387, 388, 428, 429] [389, 390, 430, 431] [391, 392, 432, 433] [393, 394, 434, 435] [395, 396, 436, 437] [399, 400, 440, 441] [401, 402, 442, 443] [403, 404, 444, 445] [405, 406, 446, 447] [407, 408, 448, 449] [409, 410, 450, 451] [411, 412, 452, 453] [413, 414, 454, 455] [415, 416, 456, 457] [417, 418, 458, 459] [419, 420, 460, 461] [421, 422, 462, 463] [423, 424, 464, 465] [425, 426, 466, 467] [427, 428, 468, 469] [429, 430, 470, 471] [431, 432, 472, 473] [433, 434, 474, 475] [435, 436, 476, 477] [437, 438, 478, 479] [439, 440, 480, 481] [441, 442, 482, 483] [443, 444, 484, 485] [445, 446, 486, 487] [447, 448, 488, 489] [449, 450, 490, 491] [451, 452, 492, 493] [453, 454, 494, 495] [455, 456, 496, 497] [457, 458, 498, 499] [459, 460, 500, 501] [461, 462, 502, 503] [463, 464, 504, 505] [465, 466, 506, 507] [467, 468, 508, 509] [469, 470, 510, 511] [471, 472, 512, 513] [473, 474, 514, 515] [475, 476, 516, 517] [477, 478, 518, 519] [481, 482] [483, 484] [485, 486] [487, 488, 521, 522] [489, 490, 523, 524] [491, 492, 525, 526] [493, 494, 527, 528] [495, 496, 529, 530] [497, 498, 531, 532] [499, 500, 533] [501, 502, 534, 535] [503, 504, 536, 537] [505, 506, 538, 539] [507, 508, 540, 541] [509, 510, 542, 543] [511, 512, 544, 545] [513, 514, 546] [515, 516] [517, 518] [519, 520] [521, 547] [522, 523, 548, 549] [524, 525, 550, 551] [526, 527, 552, 553] [528, 529, 554, 555] [530, 531, 556, 557] [532, 533, 558, 559] [534, 560] [535, 536, 561, 562] [537, 538, 563, 564] [539, 540, 565, 566] [541, 542, 567, 568] [543, 544, 569, 570] [545, 546, 571, 572] [547, 548, 573, 574] [549, 550, 575, 576] [551, 552, 577, 578] [553, 554, 579, 580] [555, 556, 581, 582] [557, 558, 583, 584] [559, 585] [560, 561, 586, 587] [562, 563, 588, 589] [564, 565, 590, 591] [566, 567, 592, 593] [568, 569, 594, 595] [570, 571, 596, 597] [572, 598] [573, 599] [574, 575, 600, 601] [576, 577, 602, 603] [578, 579, 604, 605] [580, 581, 606, 607] [582, 583, 608, 609] [584, 585, 610, 611] [586, 612] [587, 588, 613, 614] [589, 590, 615, 616] [591, 592, 617, 618] [593, 594, 619, 620] [595, 596, 621, 622] [597, 598, 623, 624] [599, 600, 625, 626] [601, 602, 627, 628] [603, 604, 629, 630] [605, 606, 631, 632] [607, 608, 633, 634] [609, 610, 635, 636] [611, 637] [612, 613, 638, 639] [614, 615, 640, 641] [616, 617, 642, 643] [618, 619, 644, 645] [620, 621, 646, 647] [622, 623, 648, 649] [624, 650] [625, 651] [626, 627, 652, 653] [628, 629, 654, 655] [630, 631, 656, 657] [632, 633, 658, 659] [634, 635, 660, 661] [636, 637, 662, 663] [638, 664] [639, 640, 665, 666] [641, 642, 667, 668] [643, 644, 669, 670] [645, 646, 671, 672] [647, 648, 673, 674] [649, 650, 675, 676]
H_Z (336 checks, sparse supports)
[0, 1, 39, 40] [2, 3, 41, 42] [4, 5, 43, 44] [6, 7, 45, 46] [8, 9, 47, 48] [10, 11, 49, 50] [12, 51] [13, 14, 52, 53] [15, 16, 54, 55] [17, 18, 56, 57] [19, 20, 58, 59] [21, 22, 60, 61] [23, 24, 62, 63] [25, 64] [26, 27, 65, 66] [28, 29, 67, 68] [30, 31, 69, 70] [32, 33, 71, 72] [34, 35, 73, 74] [36, 37, 75, 76] [38, 77] [39, 78] [40, 41, 79, 80] [42, 43, 81, 82] [44, 45, 83, 84] [46, 47, 85, 86] [48, 49, 87, 88] [50, 51, 89, 90] [52, 91] [53, 54, 92, 93] [55, 56, 94, 95] [57, 58, 96, 97] [59, 60, 98, 99] [61, 62, 100, 101] [63, 64, 102, 103] [65, 104] [66, 67, 105, 106] [68, 69, 107, 108] [70, 71, 109, 110] [72, 73, 111, 112] [74, 75, 113, 114] [76, 77, 115, 116] [78, 79, 117, 118] [80, 81, 119, 120] [82, 83, 121, 122] [84, 85, 123, 124] [86, 87, 125, 126] [88, 89, 127, 128] [90, 129] [91, 92, 130, 131] [93, 94, 132, 133] [95, 96, 134, 135] [97, 98, 136, 137] [99, 100, 138, 139] [101, 102, 140, 141] [103, 142] [104, 105, 143, 144] [106, 107, 145, 146] [108, 109, 147, 148] [110, 111, 149, 150] [112, 113, 151, 152] [114, 115, 153, 154] [116, 155] [117, 156] [118, 119, 157, 158] [120, 121, 159, 160] [122, 123, 161, 162] [124, 125, 163, 164] [126, 127, 165, 166] [128, 129, 167, 168] [130, 169] [131, 132, 170, 171] [133, 134, 172, 173] [135, 136, 174, 175] [137, 138, 176, 177] [139, 140, 178, 179] [141, 142, 180, 181] [143, 182] [144, 145, 183, 184] [146, 147, 185, 186] [148, 149, 187, 188] [150, 151, 189, 190] [152, 153, 191, 192] [154, 155, 193, 194] [156, 157, 195, 196] [158, 159, 197, 198] [160, 161, 199, 200] [162, 163, 201, 202] [164, 165, 203, 204] [166, 167, 205, 206] [168, 207] [169, 170, 208, 209] [171, 172, 210, 211] [173, 174, 212, 213] [175, 176, 214, 215] [177, 178, 216, 217] [179, 180, 218, 219] [181, 220] [182, 183, 221, 222] [184, 185, 223, 224] [186, 187, 225, 226] [188, 189, 227, 228] [190, 191, 229, 230] [192, 193, 231, 232] [194, 233] [195, 234] [196, 197, 235, 236] [198, 199, 237, 238] [200, 201, 239, 240] [202, 203, 241, 242] [204, 205, 243, 244] [206, 207, 245, 246] [208, 247, 248] [209, 210, 249, 250] [211, 212, 251, 252] [213, 214, 253, 254] [215, 216, 255, 256] [217, 218, 257, 258] [219, 220, 259, 260] [221, 261, 262] [222, 223, 263, 264] [224, 225, 265, 266] [226, 227, 267, 268] [228, 229, 269, 270] [230, 231, 271, 272] [232, 233, 273, 274] [234, 235, 275, 276] [236, 237, 277, 278] [238, 239, 279, 280] [240, 241, 281, 282] [242, 243, 283, 284] [244, 245, 285, 286] [246, 247, 287, 288] [248, 249, 289, 290] [250, 251, 291, 292] [252, 253, 293, 294] [254, 255, 295, 296] [256, 257, 297, 298] [258, 259, 299, 300] [260, 261, 301, 302] [262, 263, 303, 304] [264, 265, 305, 306] [266, 267, 307, 308] [268, 269, 309, 310] [270, 271, 311, 312] [272, 273, 313, 314] [274, 315] [275, 316] [276, 277, 317, 318] [278, 279, 319, 320] [280, 281, 321, 322] [282, 283, 323, 324] [284, 285, 325, 326] [286, 287, 327, 328] [288, 289, 329, 330] [290, 291, 331, 332] [292, 293, 333, 334] [294, 295, 335, 336] [296, 297, 337, 338] [298, 299, 339, 340] [300, 301, 341, 342] [302, 303, 343, 344] [304, 305, 345, 346] [306, 307, 347, 348] [308, 309, 349, 350] [310, 311, 351, 352] [312, 313, 353, 354] [314, 315, 355, 356] [316, 317, 357, 358] [318, 319, 359, 360] [320, 321, 361, 362] [322, 323, 363, 364] [324, 325, 365, 366] [326, 327, 367, 368] [328, 329, 369, 370] [330, 331, 371, 372] [332, 333, 373, 374] [334, 335, 375, 376] [336, 337, 377, 378] [338, 339, 379, 380] [340, 341, 381, 382] [342, 343, 383, 384] [344, 345, 385, 386] [346, 347, 387, 388] [348, 349, 389, 390] [350, 351, 391, 392] [352, 353, 393, 394] [354, 355, 395, 396] [356, 397] [357, 398] [358, 359, 399, 400] [360, 361, 401, 402] [362, 363, 403, 404] [364, 365, 405, 406] [366, 367, 407, 408] [368, 369, 409, 410] [370, 371, 411, 412] [372, 373, 413, 414] [374, 375, 415, 416] [376, 377, 417, 418] [378, 379, 419, 420] [380, 381, 421, 422] [382, 383, 423, 424] [384, 385, 425, 426] [386, 387, 427, 428] [388, 389, 429, 430] [390, 391, 431, 432] [392, 393, 433, 434] [394, 395, 435, 436] [396, 397, 437, 438] [398, 399, 439, 440] [400, 401, 441, 442] [402, 403, 443, 444] [404, 405, 445, 446] [406, 407, 447, 448] [408, 409, 449, 450] [410, 411, 451, 452] [412, 413, 453, 454] [414, 415, 455, 456] [416, 417, 457, 458] [418, 419, 459, 460] [420, 421, 461, 462] [422, 423, 463, 464] [424, 425, 465, 466] [426, 427, 467, 468] [428, 429, 469, 470] [430, 431, 471, 472] [432, 433, 473, 474] [434, 435, 475, 476] [436, 437, 477, 478] [438, 479] [439, 480] [440, 441, 481, 482] [442, 443, 483, 484] [444, 445, 485, 486] [446, 447, 487, 488] [448, 449, 489, 490] [450, 451, 491, 492] [452, 453, 493, 494] [454, 455, 495, 496] [456, 457, 497, 498] [458, 459, 499, 500] [460, 461, 501, 502] [462, 463, 503, 504] [464, 465, 505, 506] [466, 467, 507, 508] [468, 469, 509, 510] [470, 471, 511, 512] [472, 473, 513, 514] [474, 475, 515, 516] [476, 477, 517, 518] [478, 479, 519, 520] [488, 489, 522, 523] [490, 491, 524, 525] [492, 493, 526, 527] [494, 495, 528, 529] [496, 497, 530, 531] [498, 499, 532, 533] [502, 503, 535, 536] [504, 505, 537, 538] [506, 507, 539, 540] [508, 509, 541, 542] [510, 511, 543, 544] [512, 513, 545, 546] [521, 522, 547, 548] [523, 524, 549, 550] [525, 526, 551, 552] [527, 528, 553, 554] [529, 530, 555, 556] [531, 532, 557, 558] [534, 535, 560, 561] [536, 537, 562, 563] [538, 539, 564, 565] [540, 541, 566, 567] [542, 543, 568, 569] [544, 545, 570, 571] [548, 549, 574, 575] [550, 551, 576, 577] [552, 553, 578, 579] [554, 555, 580, 581] [556, 557, 582, 583] [558, 559, 584, 585] [561, 562, 587, 588] [563, 564, 589, 590] [565, 566, 591, 592] [567, 568, 593, 594] [569, 570, 595, 596] [571, 572, 597, 598] [573, 574, 599, 600] [575, 576, 601, 602] [577, 578, 603, 604] [579, 580, 605, 606] [581, 582, 607, 608] [583, 584, 609, 610] [586, 587, 612, 613] [588, 589, 614, 615] [590, 591, 616, 617] [592, 593, 618, 619] [594, 595, 620, 621] [596, 597, 622, 623] [600, 601, 626, 627] [602, 603, 628, 629] [604, 605, 630, 631] [606, 607, 632, 633] [608, 609, 634, 635] [610, 611, 636, 637] [613, 614, 639, 640] [615, 616, 641, 642] [617, 618, 643, 644] [619, 620, 645, 646] [621, 622, 647, 648] [623, 624, 649, 650] [625, 626, 651, 652] [627, 628, 653, 654] [629, 630, 655, 656] [631, 632, 657, 658] [633, 634, 659, 660] [635, 636, 661, 662] [638, 639, 664, 665] [640, 641, 666, 667] [642, 643, 668, 669] [644, 645, 670, 671] [646, 647, 672, 673] [648, 649, 674, 675] [652, 653] [654, 655] [656, 657] [658, 659] [660, 661] [662, 663] [665, 666] [667, 668] [669, 670] [671, 672] [673, 674] [675, 676]
Code ID 677-5-13 · download JSON · raw on GitHub