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[[394,13,11]] d ≤
n
394
k
13
d
11
kd²/n
3.992
w
6
X/Z
1
g
0.0236
r
3.6056
layers
2
swaps
978

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[19, 20, 48, 149, 150, 179, 278, 304, 318, 330, 370]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[96, 107, 115, 120, 144, 242, 272, 273, 290, 320, 321]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–6 (mean 5.095) · H_Z 2–6 (mean 5.083)
qubit degrees H_X 1–4 (mean 2.444) · H_Z 1–3 (mean 2.477)
trapping sets H_X (1,1)×71 (2,0)×22 (3,0)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 71 (1,2): 78 (1,3): 244 (1,4): 1 (2,0): 22 (2,1): 70 (2,2): 322 (2,3): 371 (2,4): 1358 (2,5): 8 (3,0): 14 (3,1): 230 (3,2): 677 (3,3): 2392 (3,4): 3312 (3,5): 10326 (3,6): 454 (3,7): 1494 (3,8): 7
trapping sets H_Z (1,1)×68 (2,0)×11 (3,0)×46 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 68 (1,2): 70 (1,3): 256 (2,0): 11 (2,1): 109 (2,2): 235 (2,3): 389 (2,4): 1442 (3,0): 46 (3,1): 163 (3,2): 704 (3,3): 2066 (3,4): 3347 (3,5): 10970 (3,6): 437 (3,7): 1605
witness diameter X 12.3693 · Z 10.4403 (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 = 3.606
X checkZ checkqubit site (210)2 qubits stacked (2 layers)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 978 nearest-neighbor SWAPs per round in total, at most 4 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 Open-boundary planar bivariate-bicycle code of arXiv:2504.08887, Fig. 21 family (f = x2 + x3 + y2, g = 1 + y + x2 y3; the paper's [[392,13,11]]), built by research/local2d/boundary_engine.py build_planar(15, 14, [(2, 0), (3, 0), (0, 2)], [(0, 0), (0, 1), (2, 3)]). Redundant weight>6 boundary rows are dropped by an exact rowspace test, then the engine's weight-1/decoupled cleanup is re-run; the bilayer layout is research/local2d/planar.py grid_coordinates, interaction radius 3.61.
model DeepSeek V4.1 Flash (claimed, not verified)
builds on arXiv:2504.08887
date 2026-09-28
notes Reproduces the k = 13 planar family of arXiv:2504.08887 (Fig. 21). The gate reports no exact or Weisfeiler-Leman-equivalent board entry.
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

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

[[394,13,11]] planar BB, arXiv:2504.08887 Fig. 21 ([[392,13,11]] family)

Direction & hypothesis

The local-2d-bilayer / weight-6 cell of the k = 13 band. Every advertised code of the two k = 13 planar families of arXiv:2504.08887 is absent from the board, and unlike the k = 6..12 families -- which existing bilayer entries dominate outright -- no board entry with k >= 13 reaches d >= 11 in that cell (the best is [[300,16,10]]), so a witness at d >= 11 is a record. The families are directional: the paper's [[392,13,11]] sits at (Lx, Ly) = (15, 14), not on a square.

What was searched

The two weight-6 k = 13 families of arXiv:2504.08887 (Fig. 21), built by research/local2d/boundary_engine.py build_planar at the paper's rectangular shape and at larger ones. Screening at 20k then 100k RIS trials per side (research/kit/surrogate.py), then a 3 x 1,000,000 fresh-seed confirmation pass. Squared grids and the transposed orientation were screened and discarded (see dead ends).

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 20,000 | 11 | | 100,000 | 11 | | 1,000,000 | 11 | | 1,000,000 | 11 | | 1,000,000 | 11 |

5 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,000,000 trials per side. The submitted witness reproduces it: X: d <= 11 (upper_bound), Z: d <= 11 (upper_bound).

Dead ends

  • Squared grids (the first attempt): (14,14) reads d <= 9 and the initial

    • k13-495 squares read d <= 4..9, four to five below the paper's distances. The families are directional and only reproduce at the paper's rectangular aspect ratio.

  • Wrong orientation: (14,15) reads d <= 9 where (15,14) reads 11, and (39,7)

    • reads d <= 7 where (27,10) reads 13.

  • The half-plane boundary kernel emits weight-12 boundary generators for both

    • families; they are products of weight-6 rows, and an exact rowspace test drops them, which is what puts the code in the weight-6 class.

  • Greedy weight reduction leaves empty rows (schema-rejected) and weight-1

    • rows (a qubit in no other check, giving 3..4 Tanner components); re-running the engine's weight-1/decoupled cleanup fixes both without changing k or d.

Tools

Model: DeepSeek V4.1 Flash (Zed coding agent). Repo tooling: research/local2d/boundary_engine.py, research/local2d/planar.py, research/kit/surrogate.py, verify/validate_candidate.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: RIS distance searches (numpy/OpenMP backend) on one laptop.

Reproduction

From the repo: research/local2d/boundary_engine.py build_planar(15, 14, [(2, 0), (3, 0), (0, 2)], [(0, 0), (0, 1), (2, 3)]) returns the gauge generators; drop the weight>6 rows with an exact rowspace test (rank-preserving), re-run the module's weight-1/decoupled cleanup, and place the surviving qubits on the bilayer grid with research/local2d/planar.py grid_coordinates(15, 14, kept). The polynomials are the paper's Fig. 21 bulk stabilizers.

Parity checks

X-checks 189 (max weight 6) · Z-checks 192 (max weight 6)
H_X (189 checks, sparse supports)
[13, 26, 184, 185, 215] [14, 27, 185, 186, 216] [15, 28, 186, 187, 217] [16, 29, 187, 188, 218] [17, 30, 188, 189, 219] [18, 31, 189, 190, 220] [19, 32, 190, 191, 221] [20, 33, 191, 192, 222] [21, 34, 192, 193, 223] [22, 196, 210] [23, 197, 211] [2, 26, 39, 198, 199, 229] [3, 27, 40, 199, 200, 230] [4, 28, 41, 200, 201, 231] [5, 29, 42, 201, 202, 232] [6, 30, 43, 202, 203, 233] [7, 31, 44, 203, 204, 234] [8, 32, 45, 204, 205, 235] [9, 33, 46, 205, 206, 236] [10, 34, 47, 206, 207, 237] [11, 35, 48, 207, 208, 238] [12, 36, 49, 208, 209, 239] [15, 39, 52, 212, 213, 243] [16, 40, 53, 213, 214, 244] [17, 41, 54, 214, 215, 245] [18, 42, 55, 215, 216, 246] [19, 43, 56, 216, 217, 247] [20, 44, 57, 217, 218, 248] [21, 45, 58, 218, 219, 249] [22, 46, 59, 219, 220, 250] [23, 47, 60, 220, 221, 251] [24, 48, 61, 221, 222, 252] [25, 49, 62, 222, 223, 253] [28, 52, 65, 226, 227, 257] [29, 53, 66, 227, 228, 258] [30, 54, 67, 228, 229, 259] [31, 55, 68, 229, 230, 260] [32, 56, 69, 230, 231, 261] [33, 57, 70, 231, 232, 262] [34, 58, 71, 232, 233, 263] [35, 59, 72, 233, 234, 264] [36, 60, 73, 234, 235, 265] [37, 61, 74, 235, 236, 266] [38, 62, 75, 236, 237, 267] [41, 65, 78, 240, 241, 271] [42, 66, 79, 241, 242, 272] [43, 67, 80, 242, 243, 273] [44, 68, 81, 243, 244, 274] [45, 69, 82, 244, 245, 275] [46, 70, 83, 245, 246, 276] [47, 71, 84, 246, 247, 277] [48, 72, 85, 247, 248, 278] [49, 73, 86, 248, 249, 279] [50, 74, 87, 249, 250, 280] [51, 75, 88, 250, 251, 281] [54, 78, 91, 254, 255, 285] [55, 79, 92, 255, 256, 286] [56, 80, 93, 256, 257, 287] [57, 81, 94, 257, 258, 288] [58, 82, 95, 258, 259, 289] [59, 83, 96, 259, 260, 290] [60, 84, 97, 260, 261, 291] [61, 85, 98, 261, 262, 292] [62, 86, 99, 262, 263, 293] [63, 87, 100, 263, 264, 294] [64, 88, 101, 264, 265, 295] [67, 91, 104, 268, 269, 299] [68, 92, 105, 269, 270, 300] [69, 93, 106, 270, 271, 301] [70, 94, 107, 271, 272, 302] [71, 95, 108, 272, 273, 303] [72, 96, 109, 273, 274, 304] [73, 97, 110, 274, 275, 305] [74, 98, 111, 275, 276, 306] [75, 99, 112, 276, 277, 307] [76, 100, 113, 277, 278, 308] [77, 101, 114, 278, 279, 309] [80, 104, 117, 282, 283, 313] [81, 105, 118, 283, 284, 314] [82, 106, 119, 284, 285, 315] [83, 107, 120, 285, 286, 316] [84, 108, 121, 286, 287, 317] [85, 109, 122, 287, 288, 318] [86, 110, 123, 288, 289, 319] [87, 111, 124, 289, 290, 320] [88, 112, 125, 290, 291, 321] [89, 113, 126, 291, 292, 322] [90, 114, 127, 292, 293, 323] [93, 117, 130, 296, 297, 327] [94, 118, 131, 297, 298, 328] [95, 119, 132, 298, 299, 329] [96, 120, 133, 299, 300, 330] [97, 121, 134, 300, 301, 331] [98, 122, 135, 301, 302, 332] [99, 123, 136, 302, 303, 333] [100, 124, 137, 303, 304, 334] [101, 125, 138, 304, 305, 335] [102, 126, 139, 305, 306, 336] [103, 127, 140, 306, 307, 337] [106, 130, 143, 310, 311, 341] [107, 131, 144, 311, 312, 342] [108, 132, 145, 312, 313, 343] [109, 133, 146, 313, 314, 344] [110, 134, 147, 314, 315, 345] [111, 135, 148, 315, 316, 346] [112, 136, 149, 316, 317, 347] [113, 137, 150, 317, 318, 348] [114, 138, 151, 318, 319, 349] [115, 139, 152, 319, 320, 350] [116, 140, 153, 320, 321, 351] [119, 143, 156, 324, 325, 355] [120, 144, 157, 325, 326, 356] [121, 145, 158, 326, 327, 357] [122, 146, 159, 327, 328, 358] [123, 147, 160, 328, 329, 359] [124, 148, 161, 329, 330, 360] [125, 149, 162, 330, 331, 361] [126, 150, 163, 331, 332, 362] [127, 151, 164, 332, 333, 363] [128, 152, 165, 333, 334, 364] [129, 153, 166, 334, 335, 365] [132, 156, 170, 338, 339, 369] [133, 157, 171, 339, 340, 370] [134, 158, 172, 340, 341, 371] [135, 159, 173, 341, 342, 372] [136, 160, 174, 342, 343, 373] [137, 161, 175, 343, 344, 374] [138, 162, 176, 344, 345, 375] [139, 163, 177, 345, 346, 376] [140, 164, 178, 346, 347, 377] [141, 165, 179, 347, 348, 378] [142, 166, 180, 348, 349, 379] [0, 187] [1, 188] [2, 189] [3, 190] [4, 191] [5, 192] [19, 193, 207] [7, 194] [21, 195, 209] [0, 13, 201] [1, 14, 202] [2, 15, 203] [3, 16, 204] [4, 17, 205] [5, 18, 206] [6, 193] [7, 20, 208] [8, 195] [9, 196] [10, 197] [145, 170, 352, 353, 383] [146, 171, 353, 354, 384] [147, 172, 354, 355, 385] [148, 173, 355, 356, 386] [149, 174, 356, 357, 387] [150, 175, 357, 358, 388] [151, 176, 358, 359, 389] [152, 177, 359, 360, 390] [153, 178, 360, 361, 391] [154, 179, 361, 362, 392] [155, 180, 362, 363, 393] [172, 380, 381] [173, 381, 382] [174, 382, 383] [175, 383, 384] [176, 384, 385] [177, 385, 386] [178, 386, 387] [179, 387, 388] [180, 388, 389] [181, 389, 390] [182, 390, 391] [158, 366, 367] [159, 367, 368] [160, 368, 369] [161, 369, 370] [162, 370, 371] [163, 371, 372] [164, 372, 373] [165, 373, 374] [166, 374, 375] [167, 375, 376] [168, 376, 377] [22, 35, 193, 194, 224] [23, 36, 194, 195, 225] [183, 391, 392] [169, 377, 378]
H_Z (192 checks, sparse supports)
[0, 28, 29, 187, 201, 227] [1, 29, 30, 188, 202, 228] [2, 30, 31, 189, 203, 229] [3, 31, 32, 190, 204, 230] [4, 32, 33, 191, 205, 231] [5, 33, 34, 192, 206, 232] [6, 34, 35, 193, 207, 233] [7, 35, 36, 194, 208, 234] [8, 36, 37, 195, 209, 235] [9, 37, 38, 196, 210, 236] [10, 38, 197, 211, 237] [13, 41, 42, 201, 215, 241] [14, 42, 43, 202, 216, 242] [15, 43, 44, 203, 217, 243] [16, 44, 45, 204, 218, 244] [17, 45, 46, 205, 219, 245] [18, 46, 47, 206, 220, 246] [19, 47, 48, 207, 221, 247] [20, 48, 49, 208, 222, 248] [21, 49, 50, 209, 223, 249] [22, 50, 51, 210, 224, 250] [26, 54, 55, 215, 229, 255] [27, 55, 56, 216, 230, 256] [28, 56, 57, 217, 231, 257] [29, 57, 58, 218, 232, 258] [30, 58, 59, 219, 233, 259] [31, 59, 60, 220, 234, 260] [32, 60, 61, 221, 235, 261] [33, 61, 62, 222, 236, 262] [34, 62, 63, 223, 237, 263] [35, 63, 64, 224, 238, 264] [36, 64, 225, 239, 265] [39, 67, 68, 229, 243, 269] [40, 68, 69, 230, 244, 270] [41, 69, 70, 231, 245, 271] [42, 70, 71, 232, 246, 272] [43, 71, 72, 233, 247, 273] [44, 72, 73, 234, 248, 274] [45, 73, 74, 235, 249, 275] [46, 74, 75, 236, 250, 276] [47, 75, 76, 237, 251, 277] [48, 76, 77, 238, 252, 278] [49, 77, 239, 253, 279] [52, 80, 81, 243, 257, 283] [53, 81, 82, 244, 258, 284] [54, 82, 83, 245, 259, 285] [55, 83, 84, 246, 260, 286] [56, 84, 85, 247, 261, 287] [57, 85, 86, 248, 262, 288] [58, 86, 87, 249, 263, 289] [59, 87, 88, 250, 264, 290] [60, 88, 89, 251, 265, 291] [61, 89, 90, 252, 266, 292] [65, 93, 94, 257, 271, 297] [66, 94, 95, 258, 272, 298] [67, 95, 96, 259, 273, 299] [68, 96, 97, 260, 274, 300] [69, 97, 98, 261, 275, 301] [70, 98, 99, 262, 276, 302] [71, 99, 100, 263, 277, 303] [72, 100, 101, 264, 278, 304] [73, 101, 102, 265, 279, 305] [74, 102, 103, 266, 280, 306] [78, 106, 107, 271, 285, 311] [79, 107, 108, 272, 286, 312] [80, 108, 109, 273, 287, 313] [81, 109, 110, 274, 288, 314] [82, 110, 111, 275, 289, 315] [83, 111, 112, 276, 290, 316] [84, 112, 113, 277, 291, 317] [85, 113, 114, 278, 292, 318] [86, 114, 115, 279, 293, 319] [87, 115, 116, 280, 294, 320] [91, 119, 120, 285, 299, 325] [92, 120, 121, 286, 300, 326] [93, 121, 122, 287, 301, 327] [94, 122, 123, 288, 302, 328] [95, 123, 124, 289, 303, 329] [96, 124, 125, 290, 304, 330] [97, 125, 126, 291, 305, 331] [98, 126, 127, 292, 306, 332] [99, 127, 128, 293, 307, 333] [100, 128, 129, 294, 308, 334] [104, 132, 133, 299, 313, 339] [105, 133, 134, 300, 314, 340] [106, 134, 135, 301, 315, 341] [107, 135, 136, 302, 316, 342] [108, 136, 137, 303, 317, 343] [109, 137, 138, 304, 318, 344] [110, 138, 139, 305, 319, 345] [111, 139, 140, 306, 320, 346] [112, 140, 141, 307, 321, 347] [113, 141, 142, 308, 322, 348] [114, 142, 309, 323, 349] [117, 145, 146, 313, 327, 353] [118, 146, 147, 314, 328, 354] [119, 147, 148, 315, 329, 355] [120, 148, 149, 316, 330, 356] [121, 149, 150, 317, 331, 357] [122, 150, 151, 318, 332, 358] [123, 151, 152, 319, 333, 359] [124, 152, 153, 320, 334, 360] [125, 153, 154, 321, 335, 361] [126, 154, 155, 322, 336, 362] [130, 158, 159, 327, 341, 367] [131, 159, 160, 328, 342, 368] [132, 160, 161, 329, 343, 369] [133, 161, 162, 330, 344, 370] [134, 162, 163, 331, 345, 371] [135, 163, 164, 332, 346, 372] [136, 164, 165, 333, 347, 373] [137, 165, 166, 334, 348, 374] [138, 166, 167, 335, 349, 375] [139, 167, 168, 336, 350, 376] [140, 168, 169, 337, 351, 377] [143, 172, 173, 341, 355, 381] [144, 173, 174, 342, 356, 382] [145, 174, 175, 343, 357, 383] [146, 175, 176, 344, 358, 384] [147, 176, 177, 345, 359, 385] [148, 177, 178, 346, 360, 386] [149, 178, 179, 347, 361, 387] [150, 179, 180, 348, 362, 388] [151, 180, 181, 349, 363, 389] [152, 181, 182, 350, 364, 390] [153, 182, 183, 351, 365, 391] [26, 184, 198] [39, 198, 212] [52, 212, 226] [65, 226, 240] [78, 240, 254] [91, 254, 268] [104, 268, 282] [117, 282, 296] [130, 296, 310] [143, 310, 324] [156, 324, 338] [170, 338, 352] [26, 27, 185, 199] [39, 40, 199, 213] [52, 53, 213, 227] [65, 66, 227, 241] [78, 79, 241, 255] [91, 92, 255, 269] [104, 105, 269, 283] [117, 118, 283, 297] [130, 131, 297, 311] [143, 144, 311, 325] [156, 157, 325, 339] [170, 171, 339, 353] [27, 28, 186, 200, 226] [40, 41, 200, 214, 240] [53, 54, 214, 228, 254] [66, 67, 228, 242, 268] [79, 80, 242, 256, 282] [92, 93, 256, 270, 296] [105, 106, 270, 284, 310] [118, 119, 284, 298, 324] [131, 132, 298, 312, 338] [144, 145, 312, 326, 352] [157, 158, 326, 340, 366] [171, 172, 340, 354, 380] [12, 239] [25, 253] [38, 267] [51, 281] [64, 295] [77, 309] [90, 323] [103, 337] [116, 351] [129, 365] [142, 379] [155, 393] [24, 252] [37, 266] [50, 280] [63, 294] [76, 308] [89, 322] [102, 336] [115, 350] [128, 364] [23, 51, 211, 225, 251] [62, 90, 253, 267, 293] [75, 103, 267, 281, 307] [88, 116, 281, 295, 321] [101, 129, 295, 309, 335] [127, 155, 323, 337, 363] [11, 238] [141, 169, 378] [154, 183, 392]
Code ID 394-13-11 · download JSON · raw on GitHub