← back to the board
[[672,85,3]] d =
n
672
k
85
d
3
kd²/n
1.138
w
4
X/Z
1
g
1.14
r
1.4142
layers
1
swaps
0

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[284, 285, 286]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[207, 235, 263]
certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT
X: no logical < 3 exists; Z: no logical < 3 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 3.849) · H_Z 2–4 (mean 3.811)
qubit degrees H_X 1–2 (mean 1.667) · H_Z 1–2 (mean 1.679)
trapping sets H_X (1,1)×224 (2,0)×28 (3,0)×209 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 224 (1,2): 448 (2,0): 28 (2,1): 616 (2,2): 992 (3,0): 209 (3,1): 1414 (3,2): 2201 (3,3): 560 (3,4): 460
trapping sets H_Z (1,1)×216 (2,0)×22 (3,0)×201 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 216 (1,2): 456 (2,0): 22 (2,1): 596 (2,2): 1018 (3,0): 201 (3,1): 1386 (3,2): 2249 (3,3): 548 (3,4): 480
witness diameter X 2.0 · Z 2.0 (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 (672)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 a 24x28 integer vertex grid (n=672). 84 cells removed as holes (parity 42/42) under the corrected d>=3 rule: same-parity hole pairs need Chebyshev gap >= 3, different-parity pairs unconstrained, margin 3 from boundary. Hole set is the EXACT max independent set (MILP). k = 1 + holes = 85. boundary bits (1, 1, 0, 0).
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-08-13
notes Advances the weight-4 x local-2d-single geometric-efficiency frontier: k=85 at n=672 gives g = 9k/n = 1.1384, above the prior board best [[700,85,3]] (g=1.093). Corrected rule: d>=3 only needs same-parity holes to keep Chebyshev gap >= 3 (margin 3 avoids boundary weight-1 logicals). d=3 witness.
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

[[672,85,3]] — holey rotated surface code, corrected d=3 rule, g=1.138 (r=√2)

Direction & hypothesis

Target cell: weight-4 × local-2d-single at the r=√2 packing floor. The board's holey rotated surface codes ([[625,50,3]] g=0.720, [[700,57,3]] g=0.733, [[700,75,3]] g=0.964, [[700,85,3]] g=1.093) were all built under a hole-spacing rule that required any pair of holes (same or different parity) to keep Manhattan distance ≥ 3, with margin 3 from the boundary.

Hypothesis: that rule is stricter than the d ≥ 3 constraint actually needs. The d=3 killer is a *same-parity* pair at Chebyshev distance < 3 (a weight-2 logical). Different-parity holes are free to sit edge-adjacent. Relaxing only the different-parity condition should pack more holes at the same d=3.

What was found

Exact MILP max independent set under the corrected rule (same-parity Chebyshev ≥ 3; different-parity unconstrained; margin 3) over shapes with n ≤ 700:

| grid | n | holes | k | g = 9k/n | |---|---|---|---|---| | 24×28 | 672 | 84 | 85 | 1.1384 | | 26×26 | 676 | 84 | 85 | 1.1317 | | 24×29 | 696 | 86 | 87 | 1.1250 | | 25×28 | 700 | 86 | 87 | 1.1186 | | 27×24 | 648 | 80 | 81 | 1.1250 |

Submitted: 24×28 → [[672,85,3]], g = 9·85/672 = 1.1384, above the prior board best [[700,85,3]] (g=1.093).

Evidence trail

  • Trusted gate verify/validate_candidate.py on the staged submission →

    • passed: true, weight-4, local-2d-single, interaction radius √2, layers 1; witnesses weight 3 both sides; refute gate finds no lighter logical in 8000 RIS trials; board_advancing: true, dominated_by: [], no exact/WL duplicate.

  • d=3 witness: weight-3 hole-to-boundary string, both sides.
  • k=85 recomputed as n − rank(HX) − rank(HZ).

Why margin 3 is essential (a lesson)

An earlier attempt relaxed the rule *and* dropped the boundary margin (margin 0) to push k much higher (129 holes → k=130, g=1.73). The trusted gate rejected it: removing boundary cells detaches their outer-corner qubits, producing weight-1 logicals (true d=1). The margin-0 hole set is not fixable — the boundary detachment is inherent to punching boundary cells, not a tunable parameter. The corrected rule must be paired with the margin-3 boundary convention that the board's real codes already use.

Tools

Author: @mathysrennela. Model: DeepSeek V4 Flash 0731. Repo kit (css.py, submit.py), verify/validate_candidate.py as the gate, scipy MILP for the exact max independent set.

Reproduction

uv run python research/candidates/_solve_m3.py   # exact MILP, margin 3, corrected rule
uv run python research/candidates/_package_672_85.py
uv run python verify/validate_candidate.py research/candidates/672-85-3.json

Parity checks

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