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[[288,8,15]] d ≤
n
288
k
8
d
15
kd²/n
6.25
w
8
X/Z
1
g
0.0977
r
4.0
layers
1
swaps
530

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[76, 91, 92, 106, 180, 205, 218, 219, 220, 221, 233, 237, 248, 249, 263]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[27, 40, 52, 66, 77, 78, 79, 90, 102, 103, 104, 130, 143, 147, 247]
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 6.857) · H_Z 3–8 (mean 6.857)
qubit degrees H_X 1–4 (mean 3.333) · H_Z 1–4 (mean 3.333)
trapping sets H_X (1,1)×24 (2,1)×74 (3,0)×20 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 24 (1,2): 48 (1,3): 24 (1,4): 192 (2,1): 74 (2,2): 124 (2,3): 235 (2,4): 659 (2,5): 131 (2,6): 1342 (3,0): 20 (3,1): 112 (3,2): 382 (3,3): 1176 (3,4): 2459 (3,5): 2977 (3,6): 10138 (3,7): 1773 (3,8): 13880 (3,9): 50 (3,10): 970
trapping sets H_Z (1,1)×36 (2,0)×10 (3,0)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 36 (1,2): 24 (1,3): 36 (1,4): 192 (2,0): 10 (2,1): 54 (2,2): 109 (2,3): 232 (2,4): 582 (2,5): 229 (2,6): 1338 (3,0): 10 (3,1): 103 (3,2): 415 (3,3): 999 (3,4): 2052 (3,5): 3232 (3,6): 9467 (3,7): 2627 (3,8): 13843 (3,9): 143 (3,10): 962
witness diameter X 17.72 · Z 19.4165 (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 (288)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 530 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 10 x 10 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

[[288,8,15]] single-layer weight-8 tile code (3 x 3 tile, 10 x 10 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 <= 15 on both sides.
  • Exact distance: DistQLDPC (the MaxSAT solver of arXiv:2606.12445), each CSS side posed alone with a complete

    • logical basis, returned 15 on both sides. 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)); 10 x 10 bulk with the paper's boundary. Coordinates in the submission JSON under locality.coordinates.

Parity checks

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