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[[715,19,7]] d ≤
n
715
k
19
d
7
kd²/n
1.302
w
4
X/Z
1
g
1.3
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[42, 112, 182, 259, 338, 417, 497]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[35, 36, 37, 38, 41, 109, 110]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 3.681) · H_Z 2–4 (mean 3.681)
qubit degrees H_X 1–2 (mean 1.792) · H_Z 1–2 (mean 1.792)
trapping sets H_X (1,1)×149 (2,0)×51 (3,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 149 (1,2): 566 (2,0): 51 (2,1): 305 (2,2): 1450 (3,0): 9 (3,1): 817 (3,2): 3797 (3,3): 183 (3,4): 876
trapping sets H_Z (1,1)×149 (2,0)×51 (3,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 149 (1,2): 566 (2,0): 51 (2,1): 305 (2,2): 1450 (3,0): 9 (3,1): 817 (3,2): 3797 (3,3): 183 (3,4): 876
witness diameter X 6.3246 · Z 6.0 (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 (715)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 (two-band 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=7, rows=2, m=10, 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

#[[715,19,7]] — multi-band dense-packed surface code, two-band point (d=7, rows=2, m=10, 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 = 7, rows = 2, m = 10, pitch = 6 (rows = 2 at pitch = d − 1, the published two-band packing); n = 715, k = 19 by exact rank, w = 4, single Tanner component, interaction radius √2.

  • Screening ladder: 4,000 RIS trials/side → lightest logical weight 7 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 ≤ 7 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=7, rows=2, m=10, 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 348 (max weight 4) · Z-checks 348 (max weight 4)
H_X (348 checks, sparse supports)
[0, 1] [2, 3] [4, 5] [7, 8] [9, 10] [11, 12] [14, 15] [16, 17] [18, 19] [21, 22] [23, 24] [25, 26] [28, 29] [30, 31] [32, 33] [35, 36] [37, 38] [39, 40] [42, 43] [44, 45] [46, 47] [49, 50] [51, 52] [53, 54] [56, 57] [58, 59] [60, 61] [63, 64] [65, 66] [67, 68] [1, 2, 71, 72] [3, 4, 73, 74] [5, 6, 75, 76] [8, 9, 78, 79] [10, 11, 80, 81] [12, 13, 82, 83] [15, 16, 85, 86] [17, 18, 87, 88] [19, 20, 89, 90] [22, 23, 92, 93] [24, 25, 94, 95] [26, 27, 96, 97] [29, 30, 99, 100] [31, 32, 101, 102] [33, 34, 103, 104] [36, 37, 106, 107] [38, 39, 108, 109] [40, 41, 110, 111] [43, 44, 113, 114] [45, 46, 115, 116] [47, 48, 117, 118] [50, 51, 120, 121] [52, 53, 122, 123] [54, 55, 124, 125] [57, 58, 127, 128] [59, 60, 129, 130] [61, 62, 131, 132] [64, 65, 134, 135] [66, 67, 136, 137] [68, 69, 138, 139] [70, 71, 140, 141] [72, 73, 142, 143] [74, 75, 144, 145] [77, 78, 147, 148] [79, 80, 149, 150] [81, 82, 151, 152] [84, 85, 154, 155] [86, 87, 156, 157] [88, 89, 158, 159] [91, 92, 161, 162] [93, 94, 163, 164] [95, 96, 165, 166] [98, 99, 168, 169] [100, 101, 170, 171] [102, 103, 172, 173] [105, 106, 175, 176] [107, 108, 177, 178] [109, 110, 179, 180] [112, 113, 182, 183] [114, 115, 184, 185] [116, 117, 186, 187] [119, 120, 189, 190] [121, 122, 191, 192] [123, 124, 193, 194] [126, 127, 196, 197] [128, 129, 198, 199] [130, 131, 200, 201] [133, 134, 203, 204] [135, 136, 205, 206] [137, 138, 207, 208] [141, 142, 211, 212] [143, 144, 213, 214] [145, 146, 215, 216] [148, 149, 219, 220] [150, 151, 221, 222] [152, 153, 223, 224] [155, 156, 227, 228] [157, 158, 229, 230] [159, 160, 231, 232] [162, 163, 235, 236] [164, 165, 237, 238] [166, 167, 239, 240] [169, 170, 243, 244] [171, 172, 245, 246] [173, 174, 247, 248] [176, 177, 251, 252] [178, 179, 253, 254] [180, 181, 255, 256] [183, 184, 259, 260] [185, 186, 261, 262] [187, 188, 263, 264] [190, 191, 267, 268] [192, 193, 269, 270] [194, 195, 271, 272] [197, 198, 275, 276] [199, 200, 277, 278] [201, 202, 279, 280] [204, 205, 283, 284] [206, 207, 285, 286] [208, 209, 287, 288] [210, 211, 289, 290] [212, 213, 291, 292] [214, 215, 293, 294] [216, 217, 295, 296] [218, 219, 297, 298] [220, 221, 299, 300] [222, 223, 301, 302] [224, 225, 303, 304] [226, 227, 305, 306] [228, 229, 307, 308] [230, 231, 309, 310] [232, 233, 311, 312] [234, 235, 313, 314] [236, 237, 315, 316] [238, 239, 317, 318] [240, 241, 319, 320] [242, 243, 321, 322] [244, 245, 323, 324] [246, 247, 325, 326] [248, 249, 327, 328] [250, 251, 329, 330] [252, 253, 331, 332] [254, 255, 333, 334] [256, 257, 335, 336] [258, 259, 337, 338] [260, 261, 339, 340] [262, 263, 341, 342] [264, 265, 343, 344] [266, 267, 345, 346] [268, 269, 347, 348] [270, 271, 349, 350] [272, 273, 351, 352] [274, 275, 353, 354] [276, 277, 355, 356] [278, 279, 357, 358] [280, 281, 359, 360] [282, 283, 361, 362] [284, 285, 363, 364] [286, 287, 365, 366] [290, 291, 369, 370] [292, 293, 371, 372] [294, 295, 373, 374] [296, 297, 375, 376] [298, 299, 377, 378] [300, 301, 379, 380] [302, 303, 381, 382] [304, 305, 383, 384] [306, 307, 385, 386] [308, 309, 387, 388] [310, 311, 389, 390] [312, 313, 391, 392] [314, 315, 393, 394] [316, 317, 395, 396] [318, 319, 397, 398] [320, 321, 399, 400] [322, 323, 401, 402] [324, 325, 403, 404] [326, 327, 405, 406] [328, 329, 407, 408] [330, 331, 409, 410] [332, 333, 411, 412] [334, 335, 413, 414] [336, 337, 415, 416] [338, 339, 417, 418] [340, 341, 419, 420] [342, 343, 421, 422] [344, 345, 423, 424] [346, 347, 425, 426] [348, 349, 427, 428] [350, 351, 429, 430] [352, 353, 431, 432] [354, 355, 433, 434] [356, 357, 435, 436] [358, 359, 437, 438] [360, 361, 439, 440] [362, 363, 441, 442] [364, 365, 443, 444] [366, 367, 445, 446] [368, 369, 447, 448] [370, 371, 449, 450] [372, 373, 451, 452] [374, 375, 453, 454] [376, 377, 455, 456] [378, 379, 457, 458] [380, 381, 459, 460] [382, 383, 461, 462] [384, 385, 463, 464] [386, 387, 465, 466] [388, 389, 467, 468] [390, 391, 469, 470] [392, 393, 471, 472] [394, 395, 473, 474] [396, 397, 475, 476] [398, 399, 477, 478] [400, 401, 479, 480] [402, 403, 481, 482] [404, 405, 483, 484] [406, 407, 485, 486] [408, 409, 487, 488] [410, 411, 489, 490] [412, 413, 491, 492] [414, 415, 493, 494] [416, 417, 495, 496] [418, 419, 497, 498] [420, 421, 499, 500] [422, 423, 501, 502] [424, 425, 503, 504] [426, 427, 505, 506] [428, 429, 507, 508] [430, 431, 509, 510] [432, 433, 511, 512] [434, 435, 513, 514] [436, 437, 515, 516] [438, 439, 517, 518] [440, 441, 519, 520] [442, 443, 521, 522] [444, 445, 523, 524] [448, 449] [450, 451, 526] [452, 453, 527, 528] [454, 455, 529, 530] [456, 457, 531, 532] [458, 459, 533] [460, 461, 534, 535] [462, 463, 536, 537] [464, 465, 538, 539] [466, 467, 540] [468, 469, 541, 542] [470, 471, 543, 544] [472, 473, 545, 546] [474, 475, 547] [476, 477, 548, 549] [478, 479, 550, 551] [480, 481, 552, 553] [482, 483, 554] [484, 485, 555, 556] [486, 487, 557, 558] [488, 489, 559, 560] [490, 491, 561] [492, 493, 562, 563] [494, 495, 564, 565] [496, 497, 566, 567] [498, 499, 568] [500, 501, 569, 570] [502, 503, 571, 572] [504, 505, 573, 574] [506, 507, 575] [508, 509, 576, 577] [510, 511, 578, 579] [512, 513, 580, 581] [514, 515, 582] [516, 517, 583, 584] [518, 519, 585, 586] [520, 521, 587, 588] [522, 523] [524, 525] [526, 527, 589, 590] [528, 529, 591, 592] [530, 531, 593, 594] [532, 595] [533, 534, 596, 597] [535, 536, 598, 599] [537, 538, 600, 601] [539, 602] [540, 541, 603, 604] [542, 543, 605, 606] [544, 545, 607, 608] [546, 609] [547, 548, 610, 611] [549, 550, 612, 613] [551, 552, 614, 615] [553, 616] [554, 555, 617, 618] [556, 557, 619, 620] [558, 559, 621, 622] [560, 623] [561, 562, 624, 625] [563, 564, 626, 627] [565, 566, 628, 629] [567, 630] [568, 569, 631, 632] [570, 571, 633, 634] [572, 573, 635, 636] [574, 637] [575, 576, 638, 639] [577, 578, 640, 641] [579, 580, 642, 643] [581, 644] [582, 583, 645, 646] [584, 585, 647, 648] [586, 587, 649, 650] [588, 651] [589, 652] [590, 591, 653, 654] [592, 593, 655, 656] [594, 595, 657, 658] [596, 659] [597, 598, 660, 661] [599, 600, 662, 663] [601, 602, 664, 665] [603, 666] [604, 605, 667, 668] [606, 607, 669, 670] [608, 609, 671, 672] [610, 673] [611, 612, 674, 675] [613, 614, 676, 677] [615, 616, 678, 679] [617, 680] [618, 619, 681, 682] [620, 621, 683, 684] [622, 623, 685, 686] [624, 687] [625, 626, 688, 689] [627, 628, 690, 691] [629, 630, 692, 693] [631, 694] [632, 633, 695, 696] [634, 635, 697, 698] [636, 637, 699, 700] [638, 701] [639, 640, 702, 703] [641, 642, 704, 705] [643, 644, 706, 707] [645, 708] [646, 647, 709, 710] [648, 649, 711, 712] [650, 651, 713, 714]
H_Z (348 checks, sparse supports)
[0, 1, 70, 71] [2, 3, 72, 73] [4, 5, 74, 75] [6, 76] [7, 8, 77, 78] [9, 10, 79, 80] [11, 12, 81, 82] [13, 83] [14, 15, 84, 85] [16, 17, 86, 87] [18, 19, 88, 89] [20, 90] [21, 22, 91, 92] [23, 24, 93, 94] [25, 26, 95, 96] [27, 97] [28, 29, 98, 99] [30, 31, 100, 101] [32, 33, 102, 103] [34, 104] [35, 36, 105, 106] [37, 38, 107, 108] [39, 40, 109, 110] [41, 111] [42, 43, 112, 113] [44, 45, 114, 115] [46, 47, 116, 117] [48, 118] [49, 50, 119, 120] [51, 52, 121, 122] [53, 54, 123, 124] [55, 125] [56, 57, 126, 127] [58, 59, 128, 129] [60, 61, 130, 131] [62, 132] [63, 64, 133, 134] [65, 66, 135, 136] [67, 68, 137, 138] [69, 139] [70, 140] [71, 72, 141, 142] [73, 74, 143, 144] [75, 76, 145, 146] [77, 147] [78, 79, 148, 149] [80, 81, 150, 151] [82, 83, 152, 153] [84, 154] [85, 86, 155, 156] [87, 88, 157, 158] [89, 90, 159, 160] [91, 161] [92, 93, 162, 163] [94, 95, 164, 165] [96, 97, 166, 167] [98, 168] [99, 100, 169, 170] [101, 102, 171, 172] [103, 104, 173, 174] [105, 175] [106, 107, 176, 177] [108, 109, 178, 179] [110, 111, 180, 181] [112, 182] [113, 114, 183, 184] [115, 116, 185, 186] [117, 118, 187, 188] [119, 189] [120, 121, 190, 191] [122, 123, 192, 193] [124, 125, 194, 195] [126, 196] [127, 128, 197, 198] [129, 130, 199, 200] [131, 132, 201, 202] [133, 203] [134, 135, 204, 205] [136, 137, 206, 207] [138, 139, 208, 209] [140, 141, 210, 211] [142, 143, 212, 213] [144, 145, 214, 215] [146, 216, 217] [147, 148, 218, 219] [149, 150, 220, 221] [151, 152, 222, 223] [153, 224, 225] [154, 155, 226, 227] [156, 157, 228, 229] [158, 159, 230, 231] [160, 232, 233] [161, 162, 234, 235] [163, 164, 236, 237] [165, 166, 238, 239] [167, 240, 241] [168, 169, 242, 243] [170, 171, 244, 245] [172, 173, 246, 247] [174, 248, 249] [175, 176, 250, 251] [177, 178, 252, 253] [179, 180, 254, 255] [181, 256, 257] [182, 183, 258, 259] [184, 185, 260, 261] [186, 187, 262, 263] [188, 264, 265] [189, 190, 266, 267] [191, 192, 268, 269] [193, 194, 270, 271] [195, 272, 273] [196, 197, 274, 275] [198, 199, 276, 277] [200, 201, 278, 279] [202, 280, 281] [203, 204, 282, 283] [205, 206, 284, 285] [207, 208, 286, 287] [209, 288] [210, 289] [211, 212, 290, 291] [213, 214, 292, 293] [215, 216, 294, 295] [217, 218, 296, 297] [219, 220, 298, 299] [221, 222, 300, 301] [223, 224, 302, 303] [225, 226, 304, 305] [227, 228, 306, 307] [229, 230, 308, 309] [231, 232, 310, 311] [233, 234, 312, 313] [235, 236, 314, 315] [237, 238, 316, 317] [239, 240, 318, 319] [241, 242, 320, 321] [243, 244, 322, 323] [245, 246, 324, 325] [247, 248, 326, 327] [249, 250, 328, 329] [251, 252, 330, 331] [253, 254, 332, 333] [255, 256, 334, 335] [257, 258, 336, 337] [259, 260, 338, 339] [261, 262, 340, 341] [263, 264, 342, 343] [265, 266, 344, 345] [267, 268, 346, 347] [269, 270, 348, 349] [271, 272, 350, 351] [273, 274, 352, 353] [275, 276, 354, 355] [277, 278, 356, 357] [279, 280, 358, 359] [281, 282, 360, 361] [283, 284, 362, 363] [285, 286, 364, 365] [287, 288, 366, 367] [289, 290, 368, 369] [291, 292, 370, 371] [293, 294, 372, 373] [295, 296, 374, 375] [297, 298, 376, 377] [299, 300, 378, 379] [301, 302, 380, 381] [303, 304, 382, 383] [305, 306, 384, 385] [307, 308, 386, 387] [309, 310, 388, 389] [311, 312, 390, 391] [313, 314, 392, 393] [315, 316, 394, 395] [317, 318, 396, 397] [319, 320, 398, 399] [321, 322, 400, 401] [323, 324, 402, 403] [325, 326, 404, 405] [327, 328, 406, 407] [329, 330, 408, 409] [331, 332, 410, 411] [333, 334, 412, 413] [335, 336, 414, 415] [337, 338, 416, 417] [339, 340, 418, 419] [341, 342, 420, 421] [343, 344, 422, 423] [345, 346, 424, 425] [347, 348, 426, 427] [349, 350, 428, 429] [351, 352, 430, 431] [353, 354, 432, 433] [355, 356, 434, 435] [357, 358, 436, 437] [359, 360, 438, 439] [361, 362, 440, 441] [363, 364, 442, 443] [365, 366, 444, 445] [367, 446] [368, 447] [369, 370, 448, 449] [371, 372, 450, 451] [373, 374, 452, 453] [375, 376, 454, 455] [377, 378, 456, 457] [379, 380, 458, 459] [381, 382, 460, 461] [383, 384, 462, 463] [385, 386, 464, 465] [387, 388, 466, 467] [389, 390, 468, 469] [391, 392, 470, 471] [393, 394, 472, 473] [395, 396, 474, 475] [397, 398, 476, 477] [399, 400, 478, 479] [401, 402, 480, 481] [403, 404, 482, 483] [405, 406, 484, 485] [407, 408, 486, 487] [409, 410, 488, 489] [411, 412, 490, 491] [413, 414, 492, 493] [415, 416, 494, 495] [417, 418, 496, 497] [419, 420, 498, 499] [421, 422, 500, 501] [423, 424, 502, 503] [425, 426, 504, 505] [427, 428, 506, 507] [429, 430, 508, 509] [431, 432, 510, 511] [433, 434, 512, 513] [435, 436, 514, 515] [437, 438, 516, 517] [439, 440, 518, 519] [441, 442, 520, 521] [443, 444, 522, 523] [445, 446, 524, 525] [451, 452, 526, 527] [453, 454, 528, 529] [455, 456, 530, 531] [459, 460, 533, 534] [461, 462, 535, 536] [463, 464, 537, 538] [467, 468, 540, 541] [469, 470, 542, 543] [471, 472, 544, 545] [475, 476, 547, 548] [477, 478, 549, 550] [479, 480, 551, 552] [483, 484, 554, 555] [485, 486, 556, 557] [487, 488, 558, 559] [491, 492, 561, 562] [493, 494, 563, 564] [495, 496, 565, 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Code ID 715-19-7 · download JSON · raw on GitHub