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[[361,26,3]] d ≤
n
361
k
26
d
3
kd²/n
0.648
w
4
g
0.648
r
1.4142
layers
1

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Distance

d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[88, 106, 124]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[314, 333, 351]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 (361)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

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Holey rotated surface code on integer grid L=19, period-3 hole sublattice: X plaquettes on even-parity cells, Z on odd-parity cells, X-holes at (3a,3b) even parity, Z-holes at (3a,3b) odd parity, margin 3 from boundaries (generalizes codes/625-50-3.json); standard rotated-surface boundary pairs (X chains vertical, Z chains horizontal).
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-08-09
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

[[361,26,3]] — holey rotated surface code, L=19 period-3 hole pattern

Direction & hypothesis

Target cell: weight-4 x local-2d-single, geometric efficiency g = 4kd^2/(n·rho^2·r^4). The board's best non-reference point is the holey rotated surface code [[625,50,3]] (L = 25) at g = 0.72 with r = sqrt(2), rho = 1. The hypothesis (from fieldnotes/2026-08-08-leaderboard-opportunity-map.md, Opportunity C) is that regularly spaced holes preserve the surface-code radius while adding logical qubits, and that the pattern generalizes across patch sizes. [[361,26,3]] is the L = 19 instance: same r = sqrt(2) floor, k = 26, a strictly smaller block than the L = 25 record.

What was searched

The period-3 hole sublattice was reverse-engineered from codes/625-50-3.json: X-holes at cells (3a,3b) with (i+j) even, Z-holes at cells (3a,3b) with (i+j) odd, margin 3 from every boundary, standard rotated-surface boundary pairs (X chains on the vertical edges, Z chains on the horizontal edges). The builder (research/candidates/oppC/src/c_holey.py) reproduces the board code row-for-row at L = 25 (k = 50). L = 19 (this code) and L = 25 were built; L = 31 is over the n <= 700 cap (n = 961). k tracks the hole count (9 holes on the X side at L = 19 vs 25 at L = 25, giving k = 26 vs 50).

Evidence trail

  • CSS, n = 361, k = 26, max check weight 4 (X 149 w4 + 18 w2, Z 150 w4 + 18 w2).
  • Distance: X witness weight 3, Z witness weight 3 (fresh 20,000-trials RIS

    • search); both verified in the opposite kernel and outside their own rowspace; refutation clean at 8,000 trials.

  • Locality: single layer (layers = 1), interaction radius sqrt(2) (integer

    • grid, unit plaquettes), within the local-2d-single cap of 4.0.

  • Final claim: witness-backed upper bound, d <= 3. Not exact-certified

    • (n = 361, k = 26 exceed the documented certification envelope).

  • verify/validate_candidate.py → passed: true, board_advancing: true,

    • dominated_by: [] in the weight-4 x local-2d-single cell.

Dead ends

  • L = 31 is over the n <= 700 cap (n = 961); only L = 19 and L = 25 fit.
  • A radius-audit bug: measuring check diameter by iterating dense matrix rows

    • gives a bogus r = 1.0 (rows reify as 0/1 columns); r must be measured from check supports. Corrected r = sqrt(2).

  • No layout optimization was attempted — the integer-grid layout is already at

    • the packing floor, so a re-optimized layout is not a new code (the gate rejects exact duplicates).

Tools

Model: DeepSeek V4 Flash 0731 (matches provenance.model). Repo kit (research/kit/css.py, research/kit/submit.py), verify/validate_candidate.py as the gate, cli/qldpc.py submit for witness search and PR drafting. Compute: ~1 CPU-hour for both sizes (L = 19: ~1 min build + 20k-trials RIS).

Reproduction

Hole coordinates: X-holes at (3a, 3b) with (i+j) even in [3, L-4]^2, Z-holes at (3a, 3b) with (i+j) odd; margin 3 from every boundary.

Parity checks

X-checks 167 · Z-checks 168
H_X (167 checks, sparse supports)
[0, 1, 19, 20] [2, 3, 21, 22] [4, 5, 23, 24] [6, 7, 25, 26] [8, 9, 27, 28] [10, 11, 29, 30] [12, 13, 31, 32] [14, 15, 33, 34] [16, 17, 35, 36] [20, 21, 39, 40] [22, 23, 41, 42] [24, 25, 43, 44] [26, 27, 45, 46] [28, 29, 47, 48] [30, 31, 49, 50] [32, 33, 51, 52] [34, 35, 53, 54] [36, 37, 55, 56] [38, 39, 57, 58] [40, 41, 59, 60] [42, 43, 61, 62] [44, 45, 63, 64] [46, 47, 65, 66] [48, 49, 67, 68] [50, 51, 69, 70] [52, 53, 71, 72] [54, 55, 73, 74] [58, 59, 77, 78] [62, 63, 81, 82] [64, 65, 83, 84] [68, 69, 87, 88] [70, 71, 89, 90] [74, 75, 93, 94] [76, 77, 95, 96] [78, 79, 97, 98] [80, 81, 99, 100] [82, 83, 101, 102] [84, 85, 103, 104] [86, 87, 105, 106] [88, 89, 107, 108] [90, 91, 109, 110] [92, 93, 111, 112] [96, 97, 115, 116] [98, 99, 117, 118] [100, 101, 119, 120] [102, 103, 121, 122] [104, 105, 123, 124] [106, 107, 125, 126] [108, 109, 127, 128] [110, 111, 129, 130] [112, 113, 131, 132] [114, 115, 133, 134] [116, 117, 135, 136] [118, 119, 137, 138] [122, 123, 141, 142] [124, 125, 143, 144] [128, 129, 147, 148] [130, 131, 149, 150] [134, 135, 153, 154] [136, 137, 155, 156] [138, 139, 157, 158] [140, 141, 159, 160] [142, 143, 161, 162] [144, 145, 163, 164] [146, 147, 165, 166] [148, 149, 167, 168] [150, 151, 169, 170] [152, 153, 171, 172] [154, 155, 173, 174] [156, 157, 175, 176] [158, 159, 177, 178] [160, 161, 179, 180] [162, 163, 181, 182] [164, 165, 183, 184] [166, 167, 185, 186] [168, 169, 187, 188] [172, 173, 191, 192] [176, 177, 195, 196] [178, 179, 197, 198] [182, 183, 201, 202] [184, 185, 203, 204] [188, 189, 207, 208] [190, 191, 209, 210] [192, 193, 211, 212] [194, 195, 213, 214] [196, 197, 215, 216] [198, 199, 217, 218] [200, 201, 219, 220] [202, 203, 221, 222] [204, 205, 223, 224] [206, 207, 225, 226] [210, 211, 229, 230] [212, 213, 231, 232] [214, 215, 233, 234] [216, 217, 235, 236] [218, 219, 237, 238] [220, 221, 239, 240] [222, 223, 241, 242] [224, 225, 243, 244] [226, 227, 245, 246] [228, 229, 247, 248] [230, 231, 249, 250] [232, 233, 251, 252] [236, 237, 255, 256] [238, 239, 257, 258] [242, 243, 261, 262] [244, 245, 263, 264] [248, 249, 267, 268] [250, 251, 269, 270] [252, 253, 271, 272] [254, 255, 273, 274] [256, 257, 275, 276] [258, 259, 277, 278] [260, 261, 279, 280] [262, 263, 281, 282] [264, 265, 283, 284] [266, 267, 285, 286] [268, 269, 287, 288] [270, 271, 289, 290] [272, 273, 291, 292] [274, 275, 293, 294] [276, 277, 295, 296] [278, 279, 297, 298] [280, 281, 299, 300] [282, 283, 301, 302] [286, 287, 305, 306] [290, 291, 309, 310] [292, 293, 311, 312] [296, 297, 315, 316] [298, 299, 317, 318] [302, 303, 321, 322] [304, 305, 323, 324] [306, 307, 325, 326] [308, 309, 327, 328] [310, 311, 329, 330] [312, 313, 331, 332] [314, 315, 333, 334] [316, 317, 335, 336] [318, 319, 337, 338] [320, 321, 339, 340] [324, 325, 343, 344] [326, 327, 345, 346] [328, 329, 347, 348] [330, 331, 349, 350] [332, 333, 351, 352] [334, 335, 353, 354] [336, 337, 355, 356] [338, 339, 357, 358] [340, 341, 359, 360] [19, 38] [57, 76] [95, 114] [133, 152] [171, 190] [209, 228] [247, 266] [285, 304] [323, 342] [18, 37] [56, 75] [94, 113] [132, 151] [170, 189] [208, 227] [246, 265] [284, 303] [322, 341]
H_Z (168 checks, sparse supports)
[1, 2, 20, 21] [3, 4, 22, 23] [5, 6, 24, 25] [7, 8, 26, 27] [9, 10, 28, 29] [11, 12, 30, 31] [13, 14, 32, 33] [15, 16, 34, 35] [17, 18, 36, 37] [19, 20, 38, 39] [21, 22, 40, 41] [23, 24, 42, 43] [25, 26, 44, 45] [27, 28, 46, 47] [29, 30, 48, 49] [31, 32, 50, 51] [33, 34, 52, 53] [35, 36, 54, 55] [39, 40, 58, 59] [41, 42, 60, 61] [43, 44, 62, 63] [45, 46, 64, 65] [47, 48, 66, 67] [49, 50, 68, 69] [51, 52, 70, 71] [53, 54, 72, 73] [55, 56, 74, 75] [57, 58, 76, 77] [59, 60, 78, 79] [61, 62, 80, 81] [65, 66, 84, 85] [67, 68, 86, 87] [71, 72, 90, 91] [73, 74, 92, 93] [77, 78, 96, 97] [79, 80, 98, 99] [81, 82, 100, 101] [83, 84, 102, 103] [85, 86, 104, 105] [87, 88, 106, 107] [89, 90, 108, 109] [91, 92, 110, 111] [93, 94, 112, 113] [95, 96, 114, 115] [97, 98, 116, 117] [99, 100, 118, 119] [101, 102, 120, 121] [103, 104, 122, 123] [105, 106, 124, 125] [107, 108, 126, 127] [109, 110, 128, 129] [111, 112, 130, 131] [115, 116, 134, 135] [119, 120, 138, 139] [121, 122, 140, 141] [125, 126, 144, 145] [127, 128, 146, 147] [131, 132, 150, 151] [133, 134, 152, 153] [135, 136, 154, 155] [137, 138, 156, 157] [139, 140, 158, 159] [141, 142, 160, 161] [143, 144, 162, 163] [145, 146, 164, 165] [147, 148, 166, 167] [149, 150, 168, 169] [153, 154, 172, 173] [155, 156, 174, 175] [157, 158, 176, 177] [159, 160, 178, 179] [161, 162, 180, 181] [163, 164, 182, 183] [165, 166, 184, 185] [167, 168, 186, 187] [169, 170, 188, 189] [171, 172, 190, 191] [173, 174, 192, 193] [175, 176, 194, 195] [179, 180, 198, 199] [181, 182, 200, 201] [185, 186, 204, 205] [187, 188, 206, 207] [191, 192, 210, 211] [193, 194, 212, 213] [195, 196, 214, 215] [197, 198, 216, 217] [199, 200, 218, 219] [201, 202, 220, 221] [203, 204, 222, 223] [205, 206, 224, 225] [207, 208, 226, 227] [209, 210, 228, 229] [211, 212, 230, 231] [213, 214, 232, 233] [215, 216, 234, 235] [217, 218, 236, 237] [219, 220, 238, 239] [221, 222, 240, 241] [223, 224, 242, 243] [225, 226, 244, 245] [229, 230, 248, 249] [233, 234, 252, 253] [235, 236, 254, 255] [239, 240, 258, 259] [241, 242, 260, 261] [245, 246, 264, 265] [247, 248, 266, 267] [249, 250, 268, 269] [251, 252, 270, 271] [253, 254, 272, 273] [255, 256, 274, 275] [257, 258, 276, 277] [259, 260, 278, 279] [261, 262, 280, 281] [263, 264, 282, 283] [267, 268, 286, 287] [269, 270, 288, 289] [271, 272, 290, 291] [273, 274, 292, 293] [275, 276, 294, 295] [277, 278, 296, 297] [279, 280, 298, 299] [281, 282, 300, 301] [283, 284, 302, 303] [285, 286, 304, 305] [287, 288, 306, 307] [289, 290, 308, 309] [293, 294, 312, 313] [295, 296, 314, 315] [299, 300, 318, 319] [301, 302, 320, 321] [305, 306, 324, 325] [307, 308, 326, 327] [309, 310, 328, 329] [311, 312, 330, 331] [313, 314, 332, 333] [315, 316, 334, 335] [317, 318, 336, 337] [319, 320, 338, 339] [321, 322, 340, 341] [323, 324, 342, 343] [325, 326, 344, 345] [327, 328, 346, 347] [329, 330, 348, 349] [331, 332, 350, 351] [333, 334, 352, 353] [335, 336, 354, 355] [337, 338, 356, 357] [339, 340, 358, 359] [0, 1] [2, 3] [4, 5] [6, 7] [8, 9] [10, 11] [12, 13] [14, 15] [16, 17] [343, 344] [345, 346] [347, 348] [349, 350] [351, 352] [353, 354] [355, 356] [357, 358] [359, 360]
raw submission JSON