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[[338,8,16]] d ≤
n
338
k
8
d
16
kd²/n
6.059
w
8
X/Z
1
g
0.0947
r
4.0
layers
1
swaps
627

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[65, 82, 98, 99, 102, 114, 235, 249, 250, 251, 252, 265, 269, 281, 282, 284]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[14, 65, 66, 78, 79, 91, 119, 133, 146, 161, 183, 196, 208, 221, 234, 261]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–8 (mean 6.933) · H_Z 3–8 (mean 6.933)
qubit degrees H_X 1–4 (mean 3.385) · H_Z 1–4 (mean 3.385)
trapping sets H_X (1,1)×26 (2,1)×81 (3,0)×22 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 26 (1,2): 52 (1,3): 26 (1,4): 234 (2,1): 81 (2,2): 135 (2,3): 256 (2,4): 780 (2,5): 143 (2,6): 1679 (3,0): 22 (3,1): 123 (3,2): 419 (3,3): 1289 (3,4): 2777 (3,5): 3266 (3,6): 12436 (3,7): 1949 (3,8): 17818 (3,9): 55 (3,10): 1233
trapping sets H_Z (1,1)×39 (2,0)×11 (3,0)×11 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 39 (1,2): 26 (1,3): 39 (1,4): 234 (2,0): 11 (2,1): 59 (2,2): 119 (2,3): 253 (2,4): 695 (2,5): 250 (2,6): 1675 (3,0): 11 (3,1): 113 (3,2): 456 (3,3): 1097 (3,4): 2333 (3,5): 3551 (3,6): 11683 (3,7): 2890 (3,8): 17777 (3,9): 157 (3,10): 1225
witness diameter X 16.6433 · Z 17.0 (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 = 4
X checkZ checkqubit site (338)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 627 nearest-neighbor SWAPs per round in total, at most 3 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 @fullymiddleaged
provenance submitted through the challenge
novelty novelty not audited
construction Tile code (Steffan, Choe, Breuckmann, Pereira, Eberhardt, arXiv:2504.09171) with box B = 3 on a 11 x 11 bulk, open boundaries as in the paper; X-tile {h00,h01,h11,h22,v02,v10,v20,v21}, Z-tile by their condition T2. Tile found by our exhaustive screen of 3 x 3 weight-8 tiles for the single-layer cell. Single-layer layout: each edge on its own integer site of the 45-degree rotated lattice, interaction radius 4.000.
model Claude Claude Opus 5.5 (claimed, not verified)
date 2026-10-01
family tile (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[338,8,16]] single-layer weight-8 tile code (3 x 3 tile, 11 x 11 bulk)

Direction & hypothesis

Tile codes (Steffan, Choe, Breuckmann, Pereira, Eberhardt, arXiv:2504.09171) are planar codes with translation- invariant checks and open boundaries, so a layout comes with the construction. The paper optimised k d^2 / n without a locality cap, and the board's tile entries are bilayer. We asked which tiles fit the single-layer cell (interaction radius <= 4.0, one qubit per site) when every edge of the lattice is placed on its own integer site of the 45-degree rotated lattice, and screened for codes that cell's frontier does not already beat.

What was searched

  • All 43,758 tiles of weight 8 in a 3 x 3 box (and every 2 x 2 tile, and 100,000 random 4 x 4 tiles per weight 4..8).

    • A tile was kept only if both its X-tile and its Z-tile (fixed by the paper's condition T2) fit radius 4.0.

  • The construction follows the paper: both tiles at every bulk position, physical qubits are all edges of the bulk

    • boxes, B - 1 rows of X-only and columns of Z-only boundary stabilizers truncated to those qubits. Our implementation reproduces the board's codes/578-18-20 exactly (n, k and spectral fingerprint) from that entry's stated tile.

  • Each tile was screened at one probe size, then swept over bulk sizes with aspect ratio <= 2 and n <= 400, keeping

    • codes whose (n, k, d_ub, w) no entry on the board's single-layer weight-8 frontier dominates. 82 tiles of weight 8 survived at radius exactly 4.0.

  • This X-tile, {h00, h01, h11, h22, v02, v10, v20, v21}, gives [[288,8,15]] on a 10 x 10 bulk and [[338,8,16]] on

    • 11 x 11. Its spectral fingerprint differs from every board code with the same n.

Evidence trail

  • The logical witnesses in the submission JSON show d <= 16 on both sides.
  • Exact distance: DistQLDPC (the MaxSAT solver of arXiv:2606.12445), each CSS side posed alone with a complete

    • logical basis, returned 16 on both sides (about 50 min per side). The board treats the distance as an upper bound until the maintainers certify it.

  • On the same 10 x 10 bulk the paper reports [[288,8,14]] as the best of its exhaustive 3 x 3 weight-8 search. Three

    • of our other surviving tiles give exactly [[288,8,14]] there; this one gives 15. We have not reconciled the difference with the paper's search and claim nothing about it beyond our own computation.

Dead ends

  • None of 500,000 random 4 x 4 tiles survived the single-layer screen, and no 2 x 2 weight-4 tile did.
  • 2 x 2 weight-5 and weight-6 tiles give new-looking k = 2 points such as [[242,2,11]], which has the parameters of

    • two distance-11 rotated surface-code patches; we do not submit those.

  • The screen's 100-trial distance bound overstated d at n > 300 (one bound of 19 fell to 15 under the exact solver),

    • so candidates are re-bounded with 2,000 trials before exact solving.

Tools

Claude Opus 5.5 in Claude Code; our research package (tile construction, frontier screen, layout) and DistQLDPC (github.com/guluchen/DistQLDPC); this repository's cli/qldpc.py, site/build.py (frontier) and verify/.

Reproduction

B = 3, X-tile {h00, h01, h11, h22, v02, v10, v20, v21} where h(x, y) joins vertices (x, y) and (x + 1, y) and v(x, y) joins (x, y) and (x, y + 1); Z-tile by T2 (h(x, y) <-> v(2 - x, 2 - y)); the paper's boundary, on an 11 x 11 bulk. Coordinates in the submission JSON under locality.coordinates.

Parity checks

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