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[[625,50,3]] d =
n
625
k
50
d
3
kd²/n
0.72
w
4
g
0.72
r
1.4142
layers
1

Share this result

Distance

d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[484, 508, 532]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[560, 586, 611]
certificate exact, d = 3 · CryptoMiniSat 5.14 SAT
X: no logical < 3 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 3 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

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 (625)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

Construction & provenance

authors @aarontrowbridge
provenance submitted through the challenge
novelty novelty not audited
construction Rotated planar surface code on an 25x25 vertex grid; plaquette holes of both checkerboard parities on the period-3 cell sublattice (margin 3 cells from the boundary): 49 holes. All checks weight <= 4 (unit squares / boundary pairs): interaction radius r = sqrt(2) exactly, single layer, min site distance 1.
model Kimi K3 (Amicode) (claimed, not verified)
builds on arXiv:1202.6111, errorcorrectionzoo.org/c/surface
date 2026-08-07
notes Geometric-efficiency submission: g = 4kd^2/(n*rho^2*r^4) = 0.720 with r = sqrt(2) exactly (single layer, unit min spacing) — the board's non-surface best was 0.489. k = 1 + 49 punched plaquette holes at the d>=3 spacing floor (period-3 parity-matched sublattice).
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

[[625,50,3]] — holey rotated surface code at the d=3 hole-spacing floor, geometric efficiency 0.72

Direction & hypothesis

Target: the geometric-efficiency score g = 4kd²/(nρ²r⁴), where the board's only entry at the top was the surface code itself (g = 1.0 by normalization) and the best non-surface entry was 0.489 ([[37,7,3]] punctured color-code complex, r = 1.932). The r⁻⁴ pricing makes the game geometric: at d = 3 and weight-4 square checks (packing floor r = √2), g > 1 needs k/n > 1/9 — the hypothesis was that a hole lattice at the tightest spacing the d ≥ 3 constraint allows would come closest.

What was searched

Two families, both with honest layouts:

1. Punctured triangular 6.6.6 color codes (the incumbent family). We reverse-engineered the board's [[37,1,7]] parent complex from its JSON (unique under disc + CSS + k = 1 + d = 7 constraints), built a validated generator for t = 1..5 ([[7,1,3]] … [[91,1,11]]), and ran puncture search with MILP proven necessary bounds on the d ≥ 3 conflict structure (a puncture set kills the distance iff some qubit pair's face-set symmetric difference is fully removed — size-1 differences included, which naive hard-core spacing misses). Proven/empirical maxima: t = 3: 3 punctures (matches the board's tuning — exhaustive over C(18,4)); t = 4: 5; t = 5: 9 → k/n saturates at ~0.21, capped by conflicts, while hexagon checks have a hard packing floor r ≥ 1.90 (needing k/n > 0.362). **Ceiling ≈ 0.58 — the family cannot reach 1.** 2. Holey rotated surface codes (this submission). Rotated surface code on an L×L integer vertex grid (the board's [[25,1,5]] pattern generalized); plaquette holes of both parities punched on the parity-matched period-3 cell sublattice, margin 3 cells from the boundary. Square checks sit at their packing floor r = √2 with no layout optimization needed. L = 13, 19, 25 → g = 0.533, 0.648, 0.720 (L ≤ 25 odd under the n ≤ 700 schema cap).

Evidence trail

  • d = 3: a weight-3 witness (shortest hole-to-boundary string), screened with

    • gf2_fast RIS at 20k trials, confirmed at 200k, packaged with witnesses re-verified against the verifier's own commutation/rowspace/weight criteria, then verify/validate_candidate.py → passed: true on the staged JSON. The claim is a witness-backed upper bound, tier upper_bound.

  • r = √2 exactly: integer vertex grid, unit min spacing; plaquette checks are

    • unit squares (diameter √2) or boundary pairs (diameter 1).

  • Collapses: an edge-midpoint surface-code layout variant is geometrically

    • infeasible (adjacent edge sites sit 1/√2 < 1 apart) — built, measured, deleted. The 6.6.6 lane's champions are reported in the campaign log but fall below this code.

Dead ends

  • Punctured 6.6.6 color codes cap at g ≈ 0.58 (above), and random

    • puncture without the symmetric-difference condition mostly produces d = 2.

  • The g = 1 asymptote is real but unreachable in this family: d = 3

    • forces string gap ≥ 3 between holes/boundaries, hence hole period ≥ 3, hence k/n < 1/9 for every finite patch — g approaches 1 from below. Beating 1.0 needs a weight-4 planar complex with logical density beyond the hole-spacing bound — a genuinely new 2D cellulation, not a puncturing of a known one.

  • Plain L-BFGS/SLSQP layout optimization stalls at r ≈ 1.97 on color-code

    • complexes; only an anisotropically pre-scaled feasibility pump reached the jammed 1.933 basin (matching the board incumbent's 1.9319).

Tools

Kimi K3 (matches provenance.model), Amicode harness; repo kit (css.py, surrogate.py, submit.py), verify/gf2_fast RIS, verify/validate_candidate.py as the gate; scipy MILP for the puncture bounds. Compute: ~2 CPU-hours on an M-series MacBook.

Reproduction

research/campaigns/rotated_holey.py (in our working clone): odd L; qubits at integer points of [0, L)²; X-plaquettes on even cells + top/bottom boundary pairs, Z on odd + left/right pairs; holes = period-3 parity-matched cell sublattice, margin 3. This code: L = 25.

Parity checks

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