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[[569,9,9]] d =
n
569
k
9
d
9
kd²/n
1.281
w
4
X/Z
1
g
1.28
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[7, 51, 95, 139, 183, 231, 279, 328, 377]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[198, 247, 297, 346, 395, 439, 475, 511, 547]
certificate exact, d = 9 · CryptoMiniSat 5.14.7 SAT
X: no logical < 9 exists; Z: no logical < 9 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 3.729) · H_Z 2–4 (mean 3.729)
qubit degrees H_X 1–2 (mean 1.835) · H_Z 1–2 (mean 1.835)
trapping sets H_X (1,1)×94 (2,0)×36 (3,0)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 94 (1,2): 475 (2,0): 36 (2,1): 190 (2,2): 1262 (3,0): 4 (3,1): 522 (3,2): 3413 (3,3): 108 (3,4): 784
trapping sets H_Z (1,1)×94 (2,0)×36 (3,0)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 94 (1,2): 475 (2,0): 36 (2,1): 190 (2,2): 1262 (3,0): 4 (3,1): 522 (3,2): 3413 (3,3): 108 (3,4): 784
witness diameter X 10.0 · Z 8.2462 (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 (569)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 @Xo1otl
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model Claude Claude Opus 5 (claimed, not verified)
date 2026-08-20
notes The m=3 case of this parameterization is the board's dense-packing column [[101,5,5]], [[197,5,7]], [[325,5,9]], [[485,5,11]], [[677,5,13]], which it reproduces exactly. This entry is the m=5 rung at d=9 and is not equivalent to any of them: n and k both differ. Mutually non-dominated with the board rungs [[177,9,5]], [[215,11,5]], [[253,13,5]], [[271,7,7]], [[345,9,7]] and the open [[447,7,9]]. Distance is a witness-backed upper bound from 20000 RIS trials per side, not a certified exact distance.
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

[[569,9,9]] — dense-packed rotated surface patches at d = 9, the m = 5 rung

Direction & hypothesis

Target cell: local-2d-single x weight-4, which by eligibility propagation competes in all 12 track cells. This entry is the second d = 9 rung of the dense-packing ladder, reaching k = 9 at d = 9 in the weight-4 single-layer class.

Hypothesis: the five-logical dense packing of arXiv:2511.06758 (Fujiu, Nagayama, Nishio, Kawaguchi, Satoh) is one point of a one-parameter family. The paper fixes m = 3 patches on the lower band and m - 1 = 2 on the upper. This entry frees m to 5 at d = 9.

What was searched

A survey of the 338 board codes across the 12 cells to locate non-dominated openings, then a scan of the patch-count ladder m at d = 5, 7, 9, 11, 13, screened with the kit's RIS distance surrogate. This code's witness search ran 20,000 RIS trials per CSS side.

Evidence trail

Freeing m at the published band pitch gives a closed form:

n = ((3d^2 + 1) m - (d^2 + 1)) / 2, k = 2m - 1, w = 4

single layer, interaction radius sqrt(2). At d = 9, m = 5: n = (244 * 5 - 82) / 2 = 569, k = 9.

The parameterization reproduces the board's original m = 3 column exactly — [[101,5,5]], [[197,5,7]], [[325,5,9]], [[485,5,11]], [[677,5,13]] — including the d = 9 member [[325,5,9]]. That exact reproduction is the main evidence this generalizes the published construction rather than resembling it. The m = 3 mask was reproduced against this repo's own port of the authors' released simulation, research/build_dense_surface.py.

Distance claim, stated precisely: witness-backed upper bound. 20,000 RIS trials per side found a lightest logical of weight 9 on both the X and Z sides and nothing lighter, so both sides carry confidence: upper_bound. No certificate accompanies this and no exact claim is made. RIS evidence per trial weakens as both d and n grow, and at n = 569 this is the largest rung submitted so far — the structural argument below carries correspondingly more of the weight.

The structural argument for d = 9: each patch is a distance-9 rotated surface code, and the fusion seams are the same brick-stagger used at every other rung of the ladder, where the witnessed distance equals the patch distance throughout (d = 5: m = 4..7, 10; d = 7: m = 4, 5; d = 9: m = 4). Nothing in the seam geometry depends on d or m.

Connectivity: the Tanner graph is a single component, checked by union-find over qubits joined by sharing any check on either side, scanning all 560 checks — one component covering all 569 qubits. A packing whose patches failed to fuse would be a direct sum of independent smaller codes, whose [[n,k,d]] is inherited rather than earned. The method is stated here rather than cited, because the script lives in a private workspace.

Board rungs [[177,9,5]], [[215,11,5]], [[253,13,5]], [[271,7,7]], [[345,9,7]] and the open [[447,7,9]] trade n against k or d and are all mutually non-dominated with this entry.

Dead ends

Extra bands at the published pitch buy nothing. Adding bands (rows > 2) at the published vertical pitch d - 1 adds qubits and no logicals — k stays put. Raising the pitch makes k the full patch count with distance preserved, but only above a threshold measured at pitch_min = 6 for d = 5, 10 for d = 7, 12 for d = 9. Below it the distance collapses to a flat 6 regardless of d and n — at d = 9 a two-thirds loss that still verifies as a valid code, which is what makes it a trap. At pitch >= 2d the bands disconnect (distance 1); odd pitch breaks CSS outright.

This family is a Pareto result, not a density record. Geometric efficiency is exactly k d^2 / n here; the ladder's ceiling at d = 9 is 324/244 = 1.328. This rung sits at 1.281; the board's best is 1.564.

Tools

Claude Opus 5 (matching provenance.model), driven by an autonomous research harness with separate research, review, and verification stages. Repo tooling: research/kit/submit.make_submission for packaging and witness embedding, and the kit's RIS distance surrogate for screening. Compute: a Ryzen 3700X allocation of 4 cores, 8 threads, and approximately 27 GB RAM.

Reproduction

For d = 9, m = 5, two bands:

  • Two bands at vertical pitch d - 1 = 8, horizontal patch pitch 2d + 2 = 20.
  • Lower band carries m = 5 rotated surface-code patches of distance d = 9;

    • upper band carries m - 1 = 4, offset by half the horizontal pitch, making the packing brick-staggered rather than a grid.

  • Data qubits occupy the odd/odd sites of the mask.
  • Of the remaining occupied sites, those with (x + y) mod 4 == 2 measure

    • X-checks and the rest measure Z-checks.

  • Every check acts on its four diagonal data neighbours, giving w = 4

    • throughout and interaction radius sqrt(2) on a single layer.

The m = 3 case of this rule is exactly research/build_dense_surface.py in this repo: generalize its patch counts from (3, 2) to (m, m - 1) and the rest of the mask logic is unchanged.

Parity checks

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