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[[726,38,5]] d =
n
726
k
38
d
5
kd²/n
1.309
w
4
X/Z
1
g
1.31
r
1.4142
layers
1
swaps
0

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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)
[38, 78, 119, 166, 212]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[68, 69, 105, 106, 107]
certificate exact, d = 5 · scipy/HiGHS cutoff IP
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 3.686) · H_Z 2–4 (mean 3.686)
qubit degrees H_X 1–2 (mean 1.747) · H_Z 1–2 (mean 1.747)
trapping sets H_X (1,1)×184 (2,0)×40 (3,0)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 184 (1,2): 542 (2,0): 40 (2,1): 432 (2,2): 1308 (3,0): 28 (3,1): 1132 (3,2): 3186 (3,3): 332 (3,4): 720
trapping sets H_Z (1,1)×184 (2,0)×40 (3,0)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 184 (1,2): 542 (2,0): 40 (2,1): 432 (2,2): 1308 (3,0): 28 (3,1): 1132 (3,2): 3186 (3,3): 332 (3,4): 720
witness diameter X 4.0 · Z 4.1231 (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 (726)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 Multi-band dense-packed surface code, generalized from arXiv:2511.06758 (Fujiu et al.): rows x m patches at pitch p on a shared plane (pitch_min regime); data on odd/odd sites, (x+y)%4==2 ancillas carry X-checks, each check supports its diagonal data neighbours; k by exact GF(2) rank. Layout: unit-spacing tilted lattice, interaction radius sqrt(2), single layer. Instance: d=5, rows=5, m=8, pitch=6.
builds on arXiv:2511.06758
date 2026-09-20
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

#[[726,38,5]] — multi-band dense-packed surface code, pitch_min point (d=5, rows=5, m=8, pitch=6)

Direction & hypothesis

Target cell: local-2d-single × weight-4, the geometric-efficiency cell — at this family's honest layout (tilted nearest-neighbour lattice, r = √2, single layer) the board's g = 4kd²/(nρ²r⁴) reduces to kd²/n. The submission cap was raised to admit 700 < n ≤ 1000 when w ≤ 8 and d ≤ 40; this family (w = 4, d ≤ 13) fits with room to spare, and the new band is exactly the strip where its boundary overhead thins: the 2026-09-01 campaign (fieldnotes/2026-09-01-multiband-dense-packing-method.md) measured kd²/n ≤ 1.328 under the old n ≤ 700 cap and an asymptote near 1.43 only at n ≥ 13,000. Hypothesis: fresh Pareto points at kd²/n ≈ 1.30–1.34 fill the band, on the d-columns and k-rungs the seeded packs leave open.

What was searched

A manifold re-enumeration of the multi-band dense packing (arXiv:2511.06758 generalized to rows × m patches at a free vertical pitch; builder research/multiband_surface.py, validated bit-exact against research/build_dense_surface.py at d = 3, 5, 7, 9). 138 (d, rows, m, pitch) configurations with n in (700, 1000], d ∈ {5, 7, 9, 11, 13}, in three regimes: two-band (rows = 2, pitch = d − 1), rows ≥ 3 at pitch_min(d) = 2⌊3d/4⌋, and the pitch = d + 1 k-unlock regime (kept only as an expected refutation). k by exact GF(2) rank at every point — never the closed form the campaign's section 6 caught breaking — plus CSS, single-Tanner-component, w ≤ 4 and no-empty-row filters. Board subtraction against the cell's 179 entries (interaction radius computed from the coordinates where a stored field is absent), then intra-sweep Pareto: 25 fresh rank-level points, 14 in distance-preserving regimes. Witness screen at 4,000 RIS trials per side (fast backend): 14/14 held at design distance on both sides.

Evidence trail

This code: d = 5, rows = 5, m = 8, pitch = 6 (rows ≥ 3 at pitch_min(d) = 2⌊3d/4⌋, the distance-preserving multi-band regime); n = 726, k = 38 by exact rank, w = 4, single Tanner component, interaction radius √2.

  • Screening ladder: 4,000 RIS trials/side → lightest logical weight 5 on

    • both sides, nothing lighter.

  • Packaged through the kit's own path (research/kit/submit.make_submission,

    • 8,000-trial witnesses embedded per side), then the trusted gate verify/validate_candidate.py: passed, not refuted, no exact duplicate, no WL-signature equivalent, board_advancing: true, dominated_by: [].

  • Claim, stated precisely: witness-backed upper bound, d ≤ 5 on both

    • sides. No exact claim, no certificate.

Dead ends

  • The 13 pitch = d + 1 k-unlock points (rank-level kd²/n up to 1.374) were

    • excluded without screening: the campaign's Result 3 measured distance collapse below pitch_min for rows ≥ 3 at every d ≥ 7, so their rank-only score is not evidence.

  • [[757,7,11]] (pitch_min(11) = 16) passed the gate but is dominated by the

    • board's codes/667-7-11.json (same k, same d, lower n) — not submitted.

  • d = 9 two-band produced no fresh in-band point; the d = 11 pitch_min

    • point is the dominated [[757,7,11]] above.

  • Sweep-infrastructure bug found and fixed en route: board entries storing

    • interaction_radius: null were silently dropped by the subtraction filter, understating the cell by 38 entries; the gate's own novelty check was authoritative throughout.

Tools

GLM 5.3 Flash, driven interactively in the Zed agent. Repo tooling: the validated builder research/multiband_surface.py, the kit's RIS surrogate with the gf2_fast backend, research/kit/submit.make_submission for packaging, verify/validate_candidate.py as the trusted gate. Compute: 14 gate runs (~90 min each, 7 in parallel) plus screening.

Reproduction

research/multiband_surface.py::build(d=5, rows=5, m=8, pitch=6) rebuilds (H_X, H_Z, coords) exactly. The mask rule: bands r = 0..rows−1 at y₀ = pitch·r; even bands carry m patches at x-offsets j·(2d+2), odd bands m − 1 offset d + 1; occupied sites per patch are window interiors ((x+y)%2==0 inside the (2d−1)² window), vertical edge columns at band phase (x+y)%4 ∈ {0,2}, and horizontal edge rows at the reverse phase, unioned over all patches. Data qubits sit on odd/odd sites; of the rest, (x+y)%4==2 sites measure X-checks, the rest Z-checks; each check supports its diagonal data neighbours. Coordinates are grid/2 (unit spacing), single layer, so the tilted nearest-neighbour checks span √2.

Parity checks

X-checks 344 (max weight 4) · Z-checks 344 (max weight 4)
H_X (344 checks, sparse supports)
[0, 1] [2, 3] [5, 6] [7, 8] [10, 11] [12, 13] [15, 16] [17, 18] [20, 21] [22, 23] [25, 26] [27, 28] [30, 31] [32, 33] [35, 36] [37, 38] [1, 2, 41, 42] [3, 4, 43, 44] [6, 7, 46, 47] [8, 9, 48, 49] [11, 12, 51, 52] [13, 14, 53, 54] [16, 17, 56, 57] [18, 19, 58, 59] [21, 22, 61, 62] [23, 24, 63, 64] [26, 27, 66, 67] [28, 29, 68, 69] [31, 32, 71, 72] [33, 34, 73, 74] [36, 37, 76, 77] [38, 39, 78, 79] [40, 41, 80, 81] [42, 43, 82, 83] [45, 46, 85, 86] [47, 48, 87, 88] [50, 51, 90, 91] [52, 53, 92, 93] [55, 56, 95, 96] [57, 58, 97, 98] [60, 61, 100, 101] [62, 63, 102, 103] [65, 66, 105, 106] [67, 68, 107, 108] [70, 71, 110, 111] [72, 73, 112, 113] [75, 76, 115, 116] [77, 78, 117, 118] [81, 82, 121, 122] [83, 84, 123, 124] [86, 87, 127, 128] [88, 89, 129, 130] [91, 92, 133, 134] [93, 94, 135, 136] [96, 97, 139, 140] [98, 99, 141, 142] [101, 102, 145, 146] [103, 104, 147, 148] [106, 107, 151, 152] [108, 109, 153, 154] [111, 112, 157, 158] [113, 114, 159, 160] [116, 117, 163, 164] [118, 119, 165, 166] [120, 121, 167, 168] [122, 123, 169, 170] [124, 125, 171, 172] [126, 127, 173, 174] [128, 129, 175, 176] [130, 131, 177, 178] [132, 133, 179, 180] [134, 135, 181, 182] [136, 137, 183, 184] [138, 139, 185, 186] [140, 141, 187, 188] [142, 143, 189, 190] [144, 145, 191, 192] [146, 147, 193, 194] [148, 149, 195, 196] [150, 151, 197, 198] [152, 153, 199, 200] [154, 155, 201, 202] [156, 157, 203, 204] [158, 159, 205, 206] [160, 161, 207, 208] [162, 163, 209, 210] [164, 165, 211, 212] [168, 169] [170, 171, 214, 215] [172, 173, 216, 217] [174, 175, 218] [176, 177, 219, 220] [178, 179, 221, 222] [180, 181, 223] [182, 183, 224, 225] [184, 185, 226, 227] [186, 187, 228] [188, 189, 229, 230] [190, 191, 231, 232] [192, 193, 233] [194, 195, 234, 235] [196, 197, 236, 237] [198, 199, 238] [200, 201, 239, 240] [202, 203, 241, 242] [204, 205, 243] [206, 207, 244, 245] [208, 209, 246, 247] [210, 211, 248] [212, 213] [249, 250] [214, 251, 252] [215, 216, 253, 254] [217, 218, 255, 256] [219, 257, 258] [220, 221, 259, 260] [222, 223, 261, 262] [224, 263, 264] [225, 226, 265, 266] [227, 228, 267, 268] [229, 269, 270] [230, 231, 271, 272] [232, 233, 273, 274] [234, 275, 276] [235, 236, 277, 278] [237, 238, 279, 280] [239, 281, 282] [240, 241, 283, 284] [242, 243, 285, 286] [244, 287, 288] [245, 246, 289, 290] [247, 248, 291, 292] [293, 294] [250, 251, 297, 298] [252, 253, 299, 300] [254, 255, 301, 302] [256, 257, 303, 304] [258, 259, 305, 306] [260, 261, 307, 308] [262, 263, 309, 310] [264, 265, 311, 312] [266, 267, 313, 314] [268, 269, 315, 316] [270, 271, 317, 318] [272, 273, 319, 320] [274, 275, 321, 322] [276, 277, 323, 324] [278, 279, 325, 326] [280, 281, 327, 328] [282, 283, 329, 330] [284, 285, 331, 332] [286, 287, 333, 334] [288, 289, 335, 336] [290, 291, 337, 338] [292, 293, 339, 340] [294, 295, 341, 342] [296, 297, 343, 344] [298, 299, 345, 346] [302, 303, 348, 349] [304, 305, 350, 351] [308, 309, 353, 354] [310, 311, 355, 356] [314, 315, 358, 359] [316, 317, 360, 361] [320, 321, 363, 364] [322, 323, 365, 366] [326, 327, 368, 369] [328, 329, 370, 371] [332, 333, 373, 374] [334, 335, 375, 376] [338, 339, 378, 379] [340, 341, 380, 381] [344, 345, 384, 385] [346, 347, 386, 387] [349, 350, 390, 391] [351, 352, 392, 393] [354, 355, 396, 397] [356, 357, 398, 399] [359, 360, 402, 403] [361, 362, 404, 405] [364, 365, 408, 409] [366, 367, 410, 411] [369, 370, 414, 415] [371, 372, 416, 417] [374, 375, 420, 421] [376, 377, 422, 423] [379, 380, 426, 427] [381, 382, 428, 429] [383, 384, 430, 431] [385, 386, 432, 433] [387, 388, 434, 435] [389, 390, 436, 437] [391, 392, 438, 439] [393, 394, 440, 441] [395, 396, 442, 443] [397, 398, 444, 445] [399, 400, 446, 447] [401, 402, 448, 449] [403, 404, 450, 451] [405, 406, 452, 453] [407, 408, 454, 455] [409, 410, 456, 457] [411, 412, 458, 459] [413, 414, 460, 461] [415, 416, 462, 463] [417, 418, 464, 465] [419, 420, 466, 467] [421, 422, 468, 469] [423, 424, 470, 471] [425, 426, 472, 473] [427, 428, 474, 475] [431, 432] [433, 434, 477, 478] [435, 436, 479, 480] [437, 438, 481] [439, 440, 482, 483] [441, 442, 484, 485] [443, 444, 486] [445, 446, 487, 488] [447, 448, 489, 490] [449, 450, 491] [451, 452, 492, 493] [453, 454, 494, 495] [455, 456, 496] [457, 458, 497, 498] [459, 460, 499, 500] [461, 462, 501] [463, 464, 502, 503] [465, 466, 504, 505] [467, 468, 506] [469, 470, 507, 508] [471, 472, 509, 510] [473, 474, 511] [475, 476] [512, 513] [477, 514, 515] [478, 479, 516, 517] [480, 481, 518, 519] [482, 520, 521] [483, 484, 522, 523] [485, 486, 524, 525] [487, 526, 527] [488, 489, 528, 529] [490, 491, 530, 531] [492, 532, 533] [493, 494, 534, 535] [495, 496, 536, 537] [497, 538, 539] [498, 499, 540, 541] [500, 501, 542, 543] [502, 544, 545] [503, 504, 546, 547] [505, 506, 548, 549] [507, 550, 551] [508, 509, 552, 553] [510, 511, 554, 555] [556, 557] [513, 514, 560, 561] [515, 516, 562, 563] [517, 518, 564, 565] [519, 520, 566, 567] [521, 522, 568, 569] [523, 524, 570, 571] [525, 526, 572, 573] [527, 528, 574, 575] [529, 530, 576, 577] [531, 532, 578, 579] [533, 534, 580, 581] [535, 536, 582, 583] [537, 538, 584, 585] [539, 540, 586, 587] [541, 542, 588, 589] [543, 544, 590, 591] [545, 546, 592, 593] [547, 548, 594, 595] [549, 550, 596, 597] [551, 552, 598, 599] [553, 554, 600, 601] [555, 556, 602, 603] [557, 558, 604, 605] [559, 560, 606, 607] [561, 562, 608, 609] [565, 566, 611, 612] [567, 568, 613, 614] [571, 572, 616, 617] [573, 574, 618, 619] [577, 578, 621, 622] [579, 580, 623, 624] [583, 584, 626, 627] [585, 586, 628, 629] [589, 590, 631, 632] [591, 592, 633, 634] [595, 596, 636, 637] [597, 598, 638, 639] [601, 602, 641, 642] [603, 604, 643, 644] [607, 608, 647, 648] [609, 610, 649, 650] [612, 613, 652, 653] [614, 615, 654, 655] [617, 618, 657, 658] [619, 620, 659, 660] [622, 623, 662, 663] [624, 625, 664, 665] [627, 628, 667, 668] [629, 630, 669, 670] [632, 633, 672, 673] [634, 635, 674, 675] [637, 638, 677, 678] [639, 640, 679, 680] [642, 643, 682, 683] [644, 645, 684, 685] [646, 647, 686, 687] [648, 649, 688, 689] [651, 652, 691, 692] [653, 654, 693, 694] [656, 657, 696, 697] [658, 659, 698, 699] [661, 662, 701, 702] [663, 664, 703, 704] [666, 667, 706, 707] [668, 669, 708, 709] [671, 672, 711, 712] [673, 674, 713, 714] [676, 677, 716, 717] [678, 679, 718, 719] [681, 682, 721, 722] [683, 684, 723, 724] [687, 688] [689, 690] [692, 693] [694, 695] [697, 698] [699, 700] [702, 703] [704, 705] [707, 708] [709, 710] [712, 713] [714, 715] [717, 718] [719, 720] [722, 723] [724, 725]
H_Z (344 checks, sparse supports)
[0, 1, 40, 41] [2, 3, 42, 43] [4, 44] [5, 6, 45, 46] [7, 8, 47, 48] [9, 49] [10, 11, 50, 51] [12, 13, 52, 53] [14, 54] [15, 16, 55, 56] [17, 18, 57, 58] [19, 59] [20, 21, 60, 61] [22, 23, 62, 63] [24, 64] [25, 26, 65, 66] [27, 28, 67, 68] [29, 69] [30, 31, 70, 71] [32, 33, 72, 73] [34, 74] [35, 36, 75, 76] [37, 38, 77, 78] [39, 79] [40, 80] [41, 42, 81, 82] [43, 44, 83, 84] [45, 85] [46, 47, 86, 87] [48, 49, 88, 89] [50, 90] [51, 52, 91, 92] [53, 54, 93, 94] [55, 95] [56, 57, 96, 97] [58, 59, 98, 99] [60, 100] [61, 62, 101, 102] [63, 64, 103, 104] [65, 105] [66, 67, 106, 107] [68, 69, 108, 109] [70, 110] [71, 72, 111, 112] [73, 74, 113, 114] [75, 115] [76, 77, 116, 117] [78, 79, 118, 119] [80, 81, 120, 121] [82, 83, 122, 123] [84, 124, 125] [85, 86, 126, 127] [87, 88, 128, 129] [89, 130, 131] [90, 91, 132, 133] [92, 93, 134, 135] [94, 136, 137] [95, 96, 138, 139] [97, 98, 140, 141] [99, 142, 143] [100, 101, 144, 145] [102, 103, 146, 147] [104, 148, 149] [105, 106, 150, 151] [107, 108, 152, 153] [109, 154, 155] [110, 111, 156, 157] [112, 113, 158, 159] [114, 160, 161] [115, 116, 162, 163] [117, 118, 164, 165] [119, 166] [120, 167] [121, 122, 168, 169] [123, 124, 170, 171] [125, 126, 172, 173] [127, 128, 174, 175] [129, 130, 176, 177] [131, 132, 178, 179] [133, 134, 180, 181] [135, 136, 182, 183] [137, 138, 184, 185] [139, 140, 186, 187] [141, 142, 188, 189] [143, 144, 190, 191] [145, 146, 192, 193] [147, 148, 194, 195] [149, 150, 196, 197] [151, 152, 198, 199] [153, 154, 200, 201] [155, 156, 202, 203] [157, 158, 204, 205] [159, 160, 206, 207] [161, 162, 208, 209] [163, 164, 210, 211] [165, 166, 212, 213] [171, 172, 215, 216] [173, 174, 217, 218] [177, 178, 220, 221] [179, 180, 222, 223] [183, 184, 225, 226] [185, 186, 227, 228] [189, 190, 230, 231] [191, 192, 232, 233] [195, 196, 235, 236] [197, 198, 237, 238] [201, 202, 240, 241] [203, 204, 242, 243] [207, 208, 245, 246] [209, 210, 247, 248] [214, 215, 252, 253] [216, 217, 254, 255] [219, 220, 258, 259] [221, 222, 260, 261] [224, 225, 264, 265] [226, 227, 266, 267] [229, 230, 270, 271] [231, 232, 272, 273] [234, 235, 276, 277] [236, 237, 278, 279] [239, 240, 282, 283] [241, 242, 284, 285] [244, 245, 288, 289] [246, 247, 290, 291] [249, 250, 296, 297] [251, 252, 298, 299] [253, 254, 300, 301] [255, 256, 302, 303] [257, 258, 304, 305] [259, 260, 306, 307] [261, 262, 308, 309] [263, 264, 310, 311] [265, 266, 312, 313] [267, 268, 314, 315] [269, 270, 316, 317] [271, 272, 318, 319] [273, 274, 320, 321] [275, 276, 322, 323] [277, 278, 324, 325] [279, 280, 326, 327] [281, 282, 328, 329] [283, 284, 330, 331] [285, 286, 332, 333] [287, 288, 334, 335] [289, 290, 336, 337] [291, 292, 338, 339] [293, 294, 340, 341] [295, 342] [296, 343] [297, 298, 344, 345] [299, 300, 346, 347] [301, 302, 348] [303, 304, 349, 350] [305, 306, 351, 352] [307, 308, 353] [309, 310, 354, 355] [311, 312, 356, 357] [313, 314, 358] [315, 316, 359, 360] [317, 318, 361, 362] [319, 320, 363] [321, 322, 364, 365] [323, 324, 366, 367] [325, 326, 368] [327, 328, 369, 370] [329, 330, 371, 372] [331, 332, 373] [333, 334, 374, 375] [335, 336, 376, 377] [337, 338, 378] [339, 340, 379, 380] [341, 342, 381, 382] [343, 344, 383, 384] [345, 346, 385, 386] [347, 387, 388] [348, 349, 389, 390] [350, 351, 391, 392] [352, 393, 394] [353, 354, 395, 396] [355, 356, 397, 398] [357, 399, 400] [358, 359, 401, 402] [360, 361, 403, 404] [362, 405, 406] [363, 364, 407, 408] [365, 366, 409, 410] [367, 411, 412] [368, 369, 413, 414] [370, 371, 415, 416] [372, 417, 418] [373, 374, 419, 420] [375, 376, 421, 422] [377, 423, 424] [378, 379, 425, 426] [380, 381, 427, 428] [382, 429] [383, 430] [384, 385, 431, 432] [386, 387, 433, 434] [388, 389, 435, 436] [390, 391, 437, 438] [392, 393, 439, 440] [394, 395, 441, 442] [396, 397, 443, 444] [398, 399, 445, 446] [400, 401, 447, 448] [402, 403, 449, 450] [404, 405, 451, 452] [406, 407, 453, 454] [408, 409, 455, 456] [410, 411, 457, 458] [412, 413, 459, 460] [414, 415, 461, 462] [416, 417, 463, 464] [418, 419, 465, 466] [420, 421, 467, 468] [422, 423, 469, 470] [424, 425, 471, 472] [426, 427, 473, 474] [428, 429, 475, 476] [434, 435, 478, 479] [436, 437, 480, 481] [440, 441, 483, 484] [442, 443, 485, 486] [446, 447, 488, 489] [448, 449, 490, 491] [452, 453, 493, 494] [454, 455, 495, 496] [458, 459, 498, 499] [460, 461, 500, 501] [464, 465, 503, 504] [466, 467, 505, 506] [470, 471, 508, 509] [472, 473, 510, 511] [477, 478, 515, 516] [479, 480, 517, 518] [482, 483, 521, 522] [484, 485, 523, 524] [487, 488, 527, 528] [489, 490, 529, 530] [492, 493, 533, 534] [494, 495, 535, 536] [497, 498, 539, 540] [499, 500, 541, 542] [502, 503, 545, 546] [504, 505, 547, 548] [507, 508, 551, 552] [509, 510, 553, 554] [512, 513, 559, 560] [514, 515, 561, 562] [516, 517, 563, 564] [518, 519, 565, 566] [520, 521, 567, 568] [522, 523, 569, 570] [524, 525, 571, 572] [526, 527, 573, 574] [528, 529, 575, 576] [530, 531, 577, 578] [532, 533, 579, 580] [534, 535, 581, 582] [536, 537, 583, 584] [538, 539, 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Code ID 726-38-5 · download JSON · raw on GitHub