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[[700,57,3]] d =
n
700
k
57
d
3
kd²/n
0.733
w
4
X/Z
1
g
0.733
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)
[256, 280, 304]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[494, 520, 546]
certificate exact, d = 3 · scipy/HiGHS MILP
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 3.833) · H_Z 2–4 (mean 3.85)
qubit degrees H_X 1–2 (mean 1.769) · H_Z 1–2 (mean 1.76)
trapping sets H_X (1,1)×162 (2,0)×24 (3,0)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 162 (1,2): 538 (2,0): 24 (2,1): 434 (2,2): 1345 (3,0): 66 (3,1): 1162 (3,2): 3350 (3,3): 384 (3,4): 752
trapping sets H_Z (1,1)×168 (2,0)×27 (3,0)×74 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 168 (1,2): 532 (2,0): 27 (2,1): 446 (2,2): 1327 (3,0): 74 (3,1): 1182 (3,2): 3310 (3,3): 390 (3,4): 740
witness diameter X 2.8284 · Z 2.8284 (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 (700)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 @aarontrowbridge
provenance submitted through the challenge
novelty novelty not audited
construction Rotated surface code on 28x25 vertex grid, period-3 hole lattice margin 3 (rectangular extension of [[625,50,3]])
model Muse Spark 1.2 (claimed, not verified)
date 2026-08-09
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

[[700,57,3]] — rectangular holey rotated surface code — g=0.73 (r=√2)

Construction

Rectangular 28×25 extension of the holey rotated surface code [[625,50,3]] (PR #433).

  • Grid: 28×25 vertex grid, n=700 qubits at integer points (x,y)∈[0,28)×[0,25), min spacing 1.
  • Checks: Rotated surface code plaquettes (weight ≤4, diameter √2) + boundary pairs (weight 2, diameter 1). X on even cells (i+j even), Z on odd cells.
  • Holes: Period-3 parity-matched sublattice, margin 3 cells from boundary: cells (i,j) with i%3==0, j%3==0, parity (i+j)%2==0→X-hole, ==1→Z-hole. For 28×25: 8×7=56 hole positions → 28 X-holes + 28 Z-holes. Parent k=1 + 56 holes = 57. Boundary parity adjusted for even Lx: right Z edge start flipped (0 vs 1) to preserve CSS (H_X H_Z^T=0); otherwise X bottom (1), X top (0), Z left (0) as in square case.
  • Layout: Integer grid, r=√2 exactly (unit squares √2, boundary pairs 1), layers=1, class local-2d-single, weight class weight-4.

Parameters

  • n=700, k=57, d=3 (hole-to-boundary weight-3 string), w=4, r=√2≈1.4142, layers=1
  • g = 4kd²/(n r⁴) = 9k/n (at d=3, r=√2) = 9×57/700 = 0.732857... vs 0.72 for 625-50-3.
  • Comparison: 625-50-3 (25×25) → g=0.72 (k=50); 675-50-3 (27×25) → g=0.666 (no extra hole row). 28×25 is maximal n≤700 odd/even rectangular that adds a hole row (Nx 7→8).

Verification

uv run python verify/qldpc_verify.py codes/700-57-3.json
→ css_commutation ok, k=57 matches, max_check_weight 4, distance_X/Z_witness weight=3 in_ker & nontrivial, interaction_radius 1.4142, locality_class local-2d-single, ok true

uv run python verify/validate_candidate.py <staged candidate>.json
→ passed: true, verify ok, refute false (no lighter logical in 8000 RIS trials), dedup null, novelty board_advancing true, advances weight-4 x local-2d-single board

The gate was run on the staged document, before filing, and that is the run the output above records. Run against the committed entry instead it exits 1, on the board's own dedup: the candidate is now identical to board entry codes/700-57-3.json. The qldpc_verify.py line above *does* run against the committed entry and still passes.

Claim tier upper_bound (witness-backed d≤3, distance 3 is minimal hole spacing so d=3 tight).

Reproduction

A local rectangular builder (not committed here): `build_rect(Lx=28, Ly=25, margin=3, period=3)` with CSS boundary search (brute 16 combos, picks 1,0,0,0 for 28×25). Then submit.make_submission(..., coordinates, layers=1) and validate_candidate.

Provenance

Rectangular generalization of the square L×L, margin 3, period 3 holey builder used for the uniform cases (a local script, not committed here). Even-L CSS fix derived from boundary parity scan. No external references beyond surface-code base.

Tools

Built with Amicode harness (model: Muse Spark 1.2) — research/kit + verify/gf2_fast + validate_candidate gate on mini/erlich.

Parity checks

X-checks 323 (max weight 4) · Z-checks 320 (max weight 4)
H_X (323 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] [600, 601, 625, 626] [602, 603, 627, 628] [604, 605, 629, 630] [608, 609, 633, 634] [610, 611, 635, 636] [614, 615, 639, 640] [616, 617, 641, 642] [620, 621, 645, 646] [622, 623, 647, 648] [626, 627, 651, 652] [628, 629, 653, 654] [630, 631, 655, 656] [632, 633, 657, 658] [634, 635, 659, 660] [636, 637, 661, 662] [638, 639, 663, 664] [640, 641, 665, 666] [642, 643, 667, 668] [644, 645, 669, 670] [646, 647, 671, 672] [648, 649, 673, 674] [650, 651, 675, 676] [652, 653, 677, 678] [654, 655, 679, 680] [656, 657, 681, 682] [658, 659, 683, 684] [660, 661, 685, 686] [662, 663, 687, 688] [664, 665, 689, 690] [666, 667, 691, 692] [668, 669, 693, 694] [670, 671, 695, 696] [672, 673, 697, 698] [25, 50] [75, 100] [125, 150] [175, 200] [225, 250] [275, 300] [325, 350] [375, 400] [425, 450] [475, 500] [525, 550] [575, 600] [625, 650] [24, 49] [74, 99] [124, 149] [174, 199] [224, 249] [274, 299] [324, 349] [374, 399] [424, 449] [474, 499] [524, 549] [574, 599] [624, 649] [674, 699]
H_Z (320 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] [601, 602, 626, 627] [605, 606, 630, 631] [607, 608, 632, 633] [611, 612, 636, 637] [613, 614, 638, 639] [617, 618, 642, 643] [619, 620, 644, 645] [623, 624, 648, 649] [625, 626, 650, 651] [627, 628, 652, 653] [629, 630, 654, 655] [631, 632, 656, 657] [633, 634, 658, 659] [635, 636, 660, 661] [637, 638, 662, 663] [639, 640, 664, 665] [641, 642, 666, 667] [643, 644, 668, 669] [645, 646, 670, 671] [647, 648, 672, 673] [651, 652, 676, 677] [653, 654, 678, 679] [655, 656, 680, 681] [657, 658, 682, 683] [659, 660, 684, 685] [661, 662, 686, 687] [663, 664, 688, 689] [665, 666, 690, 691] [667, 668, 692, 693] [669, 670, 694, 695] [671, 672, 696, 697] [673, 674, 698, 699] [0, 1] [2, 3] [4, 5] [6, 7] [8, 9] [10, 11] [12, 13] [14, 15] [16, 17] [18, 19] [20, 21] [22, 23] [675, 676] [677, 678] [679, 680] [681, 682] [683, 684] [685, 686] [687, 688] [689, 690] [691, 692] [693, 694] [695, 696] [697, 698]
Code ID 700-57-3 · download JSON · raw on GitHub