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[[625,33,5]] d =
n
625
k
33
d
5
kd²/n
1.32
w
4
X/Z
1
g
1.32
r
1.4142
layers
1
swaps
0

Share this result

Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[590, 593, 594, 606, 607]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[554, 574, 592, 607, 622]
certificate exact, d = 5 · CryptoMiniSat 5.14.7 SAT
X: no logical < 5 exists; Z: no logical < 5 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 3.699) · H_Z 2–4 (mean 3.699)
qubit degrees H_X 1–2 (mean 1.752) · H_Z 1–2 (mean 1.752)
trapping sets H_X (1,1)×155 (2,0)×32 (3,0)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 155 (1,2): 470 (2,0): 32 (2,1): 369 (2,2): 1140 (3,0): 25 (3,1): 968 (3,2): 2788 (3,3): 289 (3,4): 628
trapping sets H_Z (1,1)×155 (2,0)×32 (3,0)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 155 (1,2): 470 (2,0): 32 (2,1): 369 (2,2): 1140 (3,0): 25 (3,1): 968 (3,2): 2788 (3,3): 289 (3,4): 628
witness diameter X 4.0 · Z 4.0 (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 (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
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

[[625,33,5]] — multi-band dense packing at d = 5, a Pareto point between the ladder and the family's best

Direction & hypothesis

Target cell: local-2d-single × weight-4. The d = 5 region between the two-band ladder rungs (k odd, notes/367-19-5.md) and the family's best multi-band point codes/676-36-5.json (k = 36) had no board entry at k = 32–35 below n = 659. Hypothesis: the multi-band (rows, m) manifold holds an undominated point there.

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 = pitch_min(5) = 6 was enumerated under n ≤ 700 (77 points) and subtracted against the board. This code (rows = 6, m = 6) was the highest-g undominated survivor after the family's better points ([[659,35,5]], submitted separately; codes/676-36-5.json). 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 = 6 bands, m = 6 patches per even band (5 per odd band), vertical pitch 6, horizontal patch pitch 2d + 2 = 12, d = 5: k = 6·6 − 3 = 33 (the full patch count), n = 625, 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 5 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: no board entry dominates this code — the ladder rungs at d = 5 top out at k = 35/n = 671 (PR #745) and the family's best is k = 36/n = 676; this entry's k = 33 at n = 625 is mutually non-dominated with both.

Dead ends

  • **Higher-g uncovered d = 5 manifold points exist but are dominated or

    • near-neighbours**: [[659,35,5]] (g = 1.328) dominates this code's neighbours at k = 35 and was submitted separately; points above g = 1.32 at k ≤ 34 do not exist in the manifold under the cap.

  • rows = 8, m = 6 (n = 820) exceeds the cap; the family's next d = 5 step

    • is off-board-legal territory.

  • The two-band raised-pitch curve is dominated by the two-band ladder

    • (same k = 2m − 1, strictly more n) — confirmed across the manifold.

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 = 5, rows = 6, m = 6, pitch = 6:

  • Six bands at vertical pitch 6, horizontal patch pitch 2d + 2 = 12. Even

    • bands (r = 0, 2, 4) carry m = 6 distance-5 rotated-surface-code patches at x-offsets j·12; odd bands (r = 1, 3, 5) carry m − 1 = 5, offset half a horizontal pitch (6) to the right. Total 33 patches, k = 33.

  • Occupied sites, per band r at y₀ = 6r with patch centers x₀ ∈ {12j} (even

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