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[[578,14,7]] d =
n
578
k
14
d
7
kd²/n
1.187
w
4
X/Z
1
g
1.19
r
1.4142
layers
1
swaps
0

Share this result

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)
[73, 95, 118, 132, 154, 176, 197]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[135, 151, 156, 174, 175, 176, 177]
certificate exact, d = 7 · CryptoMiniSat 5.14.7 SAT
X: no logical < 7 exists; Z: no logical < 7 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 3.691) · H_Z 2–4 (mean 3.691)
qubit degrees H_X 1–2 (mean 1.801) · H_Z 1–2 (mean 1.801)
trapping sets H_X (1,1)×115 (2,0)×39 (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): 115 (1,2): 463 (2,0): 39 (2,1): 247 (2,2): 1184 (3,0): 9 (3,1): 683 (3,2): 3064 (3,3): 159 (3,4): 708
trapping sets H_Z (1,1)×115 (2,0)×39 (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): 115 (1,2): 463 (2,0): 39 (2,1): 247 (2,2): 1184 (3,0): 9 (3,1): 683 (3,2): 3064 (3,3): 159 (3,4): 708
witness diameter X 6.0 · Z 6.0828 (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 (578)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

[[578,14,7]] — multi-band dense packing at d = 7, filling the k-gap between the family's board points

Direction & hypothesis

Target cell: local-2d-single × weight-4. At d = 7 the multi-band family's board points are [[418,10,7]] (rows=4, m=3) and [[615,15,7]] (rows=6, m=3); the (rows, m) grid between and beyond them was never enumerated. Hypothesis: the manifold holds a k = 14 point below the n = 615 of the k = 15 entry — a Pareto opening neither neighbour covers.

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(7) = 10 was enumerated under n ≤ 700: 22 points, 20 uncovered, of which this code is the highest-g undominated survivor (g = 14·49/578 ≈ 1.1869). 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 = 4 bands, m = 4 patches per even band (3 per odd band), vertical pitch 10, horizontal patch pitch 2d + 2 = 16, d = 7: k = 4·4 − 2 = 14 (the full patch count), n = 578, 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 7 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 — [[418,10,7]] has less k, [[615,15,7]] and the ladder rung [[641,17,7]] have more k but more n. This entry is mutually non-dominated with all of them.

Dead ends

  • **The best uncovered d = 7 point by g ([[578,14,7]] at 1.1869) is below the

    • best covered one ([[615,15,7]] at 1.1951)** — the family's d = 7 frontier under the cap is already placed; this entry adds a Pareto point, not a frontier advance.

  • rows ≥ 5 at m ≥ 4 exceeds the cap (n > 700); the next family step at

    • d = 7 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) at every d — confirmed across the manifold, not worth submitting anywhere.

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 = 7, rows = 4, m = 4, pitch = 10:

  • Four bands at vertical pitch 10, horizontal patch pitch 2d + 2 = 16. Even

    • bands (r = 0, 2) carry m = 4 distance-7 rotated-surface-code patches at x-offsets j·16; odd bands (r = 1, 3) carry m − 1 = 3, offset half a horizontal pitch (8) to the right. Total 14 patches, k = 14.

  • Occupied sites, per band r at y₀ = 10r with patch centers x₀ ∈ {16j} (even

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