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[[663,91,3]] d =
n
663
k
91
d
3
kd²/n
1.235
w
4
X/Z
1
g
1.24
r
1.4142
layers
1
swaps
0

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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)
[258, 282, 307]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[616, 639, 660]
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 12 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.545) · H_Z 2–4 (mean 3.545)
qubit degrees H_X 1–2 (mean 1.529) · H_Z 1–2 (mean 1.529)
trapping sets H_X (1,1)×312 (2,0)×26 (3,0)×404 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 312 (1,2): 351 (2,0): 26 (2,1): 754 (2,2): 572 (3,0): 404 (3,1): 1213 (3,2): 970 (3,3): 598 (3,4): 156
trapping sets H_Z (1,1)×312 (2,0)×26 (3,0)×404 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 312 (1,2): 351 (2,0): 26 (2,1): 754 (2,2): 572 (3,0): 404 (3,1): 1200 (3,2): 983 (3,3): 598 (3,4): 156
witness diameter X 2.2361 · 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 (663)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 contributed via qldpc submit
model GLM 5.3 Flash (claimed, not verified)
date 2026-08-29
family other (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

[[663,91,3]] — multi-band dense packing at d = 3, the family's highest logical count

Direction & hypothesis

Target cell: local-2d-single × weight-4. The board's d = 3 multi-band entries (codes/398-54-3.json, codes/570-78-3.json, codes/672-85-3.json, codes/700-85-3.json) hold k up to 85 at n ≥ 672. Hypothesis: the family's (rows, m) manifold under the n ≤ 700 cap still holds a higher-k point that dominates them — the manifold had never been enumerated exhaustively, only sampled.

What was searched

A manifold subtraction, not a point search. The multi-band family's mask was reconstructed as a per-band union rule (window interiors at (x+y)%2==0; vertical edge columns at band phase (x+y)%4∈{0,2}; horizontal edge rows at the reverse phase) and validated bit-exactly against research/build_dense_surface.py (the rows=2, pitch=d−1, m=3 case) at d = 3, 5, 7, and against the exact (n, k) of every board member of the family — 15/15 reproduced, each CSS-commuting, single Tanner component, max weight 4. The full (rows, m) manifold at pitch = 4 (the board's d = 3 operating pitch; the threshold was never measured at d = 3) was enumerated under n ≤ 700: 263 points, 261 uncovered, of which this code is the highest-g survivor (g = 91·9/663 ≈ 1.2353). Screening used the kit's RIS surrogate; this code's witness search ran 20,000 RIS trials per CSS side.

Evidence trail

The submitted code is rows = 14 bands, m = 7 patches per even band (6 per odd band), vertical pitch 4, horizontal patch pitch 2d + 2 = 8, d = 3: k = 14·7 − 7 = 91 (the full patch count), n = 663, w = 4, single layer, interaction radius √2.

Distance claim, stated precisely: witness-backed upper bound. 20,000 RIS trials per side found a lightest logical of weight 3 on both the X and Z sides and nothing lighter, so both sides carry confidence: upper_bound. No certificate and no exact claim. The trusted gate (verify/validate_candidate.py) passed with board_advancing: true and no dominating board entry.

Domination, stated precisely: this code dominates codes/672-85-3.json and codes/700-85-3.json (higher k, lower n, same d). It is mutually non-dominated with codes/570-78-3.json (k 78 < 91, n 570 < 663) and with the d = 3 record codes/656-114-3.json (different mechanism: chamfer/checkerboard, g = 1.564 — this entry is a parity-club and Pareto submission, not a record chase).

Dead ends

  • **The remaining 260 uncovered d = 3 manifold points are all dominated or

    • near-neighbours** of this code or of board entries; per the refutation-budget policy they were left unstaged.

  • The family's d = 3 asymptote is g → 4·9/(3·9+1) = 1.286 as rows → ∞;

    • under the n ≤ 700 cap the reachable maximum is this code's 1.235. The chamfer family's 1.564 record is out of this mechanism's reach — closing that gap needs the chamfer-d ≥ 4 generalization or the d = 3 SAT program, not more manifold points.

  • pitch_min(3) is unmeasured. The board's d = 3 entries all use pitch 4;

    • whether pitch 2 (the d − 1 analog) collapses distance the way sub-threshold pitches do at d = 5/7/9 was not tested.

Tools

GLM 5.3 Flash (matching provenance.model), driven interactively in the Zed agent. Repo tooling: the kit's RIS surrogate for screening and witness search (research/kit/surrogate.py, research/kit/submit.make_submission for packaging), verify/validate_candidate.py as the trusted gate. The manifold enumeration and subtraction were run with the validated builder described above; the mask rule is fully specified in Reproduction so the builder can be rewritten from this note alone.

Reproduction

For d = 3, rows = 14, m = 7, pitch = 4:

  • Fourteen bands at vertical pitch 4, horizontal patch pitch 2d + 2 = 8. Even

    • bands (r = 0, 2, …, 12) carry m = 7 distance-3 rotated-surface-code patches at x-offsets j·8; odd bands carry m − 1 = 6, offset half a horizontal pitch (4) to the right. Total 91 patches, k = 91.

  • Occupied sites, per band r at y₀ = 4r with patch centers x₀ ∈ {8j} (even

    • r) or {4 + 8j} (odd r): (a) window interiors [x₀+1, x₀+2d−1] × [y₀+1, y₀+2d−1] with (x+y) % 2 == 0; (b) vertical edge columns {x₀, x₀+2d} × rows [y₀+2, y₀+2d−2] with (x+y) % 4 == 0 (even r) or 2 (odd r); (c) horizontal edge rows {y₀, y₀+2d} × columns [x₀+2, x₀+2d−1] with (x+y) % 4 == 2 (even r) or 0 (odd r). Union over all bands.

  • Data qubits occupy the odd/odd sites of the mask. Of the remaining

    • occupied sites, those with (x + y) % 4 == 2 measure X-checks and the rest measure Z-checks; every check acts on its four diagonal data neighbours.

  • The rows = 2, pitch = d − 1, m = 3 case of this rule is exactly

    • research/build_dense_surface.py in this repo.

Parity checks

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