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[[364,13,11]] d ≤
n
364
k
13
d
11
kd²/n
4.321
w
6
X/Z
1
g
0.0256
r
3.6056
layers
2
swaps
911

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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)
[15, 103, 104, 132, 205, 243, 269, 282, 295, 335, 355]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[66, 70, 74, 87, 111, 224, 246, 253, 254, 255, 260]
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 5.134) · H_Z 2–6 (mean 5.061)
qubit degrees H_X 1–3 (mean 2.426) · H_Z 1–3 (mean 2.489)
trapping sets H_X (1,1)×67 (2,0)×21 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 67 (1,2): 75 (1,3): 222 (2,0): 21 (2,1): 77 (2,2): 290 (2,3): 389 (2,4): 1202 (3,0): 12 (3,1): 213 (3,2): 731 (3,3): 2182 (3,4): 3377 (3,5): 8998 (3,6): 409 (3,7): 1284
trapping sets H_Z (1,1)×54 (2,0)×12 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 54 (1,2): 78 (1,3): 232 (2,0): 12 (2,1): 92 (2,2): 214 (2,3): 417 (2,4): 1280 (3,0): 30 (3,1): 143 (3,2): 653 (3,3): 1878 (3,4): 3535 (3,5): 9589 (3,6): 464 (3,7): 1405
witness diameter X 14.4222 · Z 11.7047 (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 (198)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 911 nearest-neighbor SWAPs per round in total, at most 5 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)]), then reduced from n = 394 to n = 364 by 26 restricted r = 1 lattice grafts (arXiv:2504.08887 Sec. III.5; the move set of research/local2d/boundary_engine.py graft_r1_safe) at distance floor 11 plus the engine's weight-1/decoupled cleanup; bilayer layout research/local2d/planar.py grid_coordinates, interaction radius 3.61.
model DeepSeek V4.1 Flash (claimed, not verified)
builds on arXiv:2504.08887
date 2026-09-29
notes Grafted from the k = 13 planar family of arXiv:2504.08887 (Fig. 21); the paper's own grafted code is [[362,13,11]]. 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

[[364,13,11]] grafted planar BB, arXiv:2504.08887 Fig. 21 family

Direction & hypothesis

The local-2d-bilayer / weight-6 cell of the k = 13 band, one rung below the ungrafted family head. The paper's own grafted code [[362,13,11]] is absent from the board, and no board entry with k >= 13 reaches d >= 11 in that cell, so a witness at d >= 11 is a record. Lattice grafting (arXiv:2504.08887 Sec. III.5) removes boundary qubits at fixed k, weight and distance, so it is the cheapest way to move the family head toward the paper's n = 362.

What was searched

The k13-392 family head at (Lx, Ly) = (15, 14), built by research/local2d/boundary_engine.py build_planar, then reduced by restricted r = 1 lattice grafts (a qubit in exactly one stabilizer of some type, removed with that stabilizer) at distance floor 11. Removal order seeds 3, 5 and 7 were tried; 3 and 7 both stall at n = 368 and 5 at n = 375. Screening at 5k RIS trials per removal, confirmed at 40k with a third seed on every accepted block of three, then a 20k / 100k / 3 x 1,000,000 fresh-seed ladder on the result.

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

  • The graft move can turn an opposite-type row of weight 2 into a weight-1

    • stabilizer, leaving a decoupled qubit: the first grafted result (n = 368) split into a 364-qubit block plus four singletons and the verifier rejected it as a direct sum. Re-running the engine's cleanup removes the four qubits and reconnects the code at n = 364.

  • The restricted r = 1 move set saturates: seeds 3 and 7 both stall at n = 368

    • before cleanup (30 qubits removed from n = 394), two short of the paper's n = 362, whose raw family head is two qubits smaller.

  • Squared grids and the transposed orientation of the family head read 4..5

    • below the paper's distances and were discarded (see the ungrafted note).

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)]) gives the family head; drop the weight>6 rows with an exact rowspace test, then apply restricted r = 1 lattice grafts at distance floor 11 (the move set of the module's graft_r1_safe) and re-run the module's weight-1/decoupled cleanup; 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 172 (max weight 6) · Z-checks 179 (max weight 6)
H_X (172 checks, sparse supports)
[9, 21, 174, 175, 199] [10, 22, 175, 176, 200] [11, 23, 176, 177, 201] [12, 24, 177, 178, 202] [13, 25, 178, 179, 203] [14, 26, 179, 180, 204] [15, 27, 180, 181, 205] [16, 28, 181, 206] [2, 20, 33, 183, 184, 212] [3, 21, 34, 184, 185, 213] [4, 22, 35, 185, 186, 214] [5, 23, 36, 186, 187, 215] [24, 37, 187, 188, 216] [6, 25, 38, 188, 189, 217] [26, 39, 189, 190, 218] [7, 27, 40, 190, 191, 219] [28, 41, 191, 192, 220] [29, 42, 192, 193, 221] [30, 43, 193, 194, 222] [10, 33, 46, 195, 196, 225] [11, 34, 47, 196, 197, 226] [12, 35, 48, 197, 198, 227] [13, 36, 49, 198, 199, 228] [14, 37, 50, 199, 200, 229] [15, 38, 51, 200, 201, 230] [16, 39, 52, 201, 202, 231] [17, 40, 53, 202, 203, 232] [18, 41, 54, 203, 204, 233] [19, 42, 55, 204, 205, 234] [22, 46, 58, 209, 210, 238] [23, 47, 59, 210, 211, 239] [24, 48, 60, 211, 212, 240] [25, 49, 61, 212, 213, 241] [26, 50, 62, 213, 214, 242] [27, 51, 63, 214, 215, 243] [28, 52, 64, 215, 216, 244] [29, 53, 65, 216, 217, 245] [30, 54, 66, 217, 218, 246] [31, 55, 67, 218, 219, 247] [32, 56, 68, 219, 220, 248] [36, 59, 72, 223, 224, 253] [37, 60, 73, 224, 225, 254] [38, 61, 74, 225, 226, 255] [39, 62, 75, 226, 227, 256] [40, 63, 76, 227, 228, 257] [41, 64, 77, 228, 229, 258] [42, 65, 78, 229, 230, 259] [43, 66, 79, 230, 231, 260] [44, 67, 80, 231, 232, 261] [45, 68, 81, 232, 233, 262] [48, 71, 84, 235, 236, 266] [49, 72, 85, 236, 237, 267] [50, 73, 86, 237, 238, 268] [51, 74, 87, 238, 239, 269] [52, 75, 88, 239, 240, 270] [53, 76, 89, 240, 241, 271] [54, 77, 90, 241, 242, 272] [55, 78, 91, 242, 243, 273] [56, 79, 92, 243, 244, 274] [57, 80, 93, 244, 245, 275] [60, 84, 97, 249, 250, 279] [61, 85, 98, 250, 251, 280] [62, 86, 99, 251, 252, 281] [63, 87, 100, 252, 253, 282] [64, 88, 101, 253, 254, 283] [65, 89, 102, 254, 255, 284] [66, 90, 103, 255, 256, 285] [67, 91, 104, 256, 257, 286] [68, 92, 105, 257, 258, 287] [69, 93, 106, 258, 259, 288] [70, 94, 107, 259, 260, 289] [73, 97, 110, 263, 264, 292] [74, 98, 111, 264, 265, 293] [75, 99, 112, 265, 266, 294] [76, 100, 113, 266, 267, 295] [77, 101, 114, 267, 268, 296] [78, 102, 115, 268, 269, 297] [79, 103, 116, 269, 270, 298] [80, 104, 117, 270, 271, 299] [81, 105, 118, 271, 272, 300] [82, 106, 119, 272, 273, 301] [83, 107, 120, 273, 274, 302] [86, 110, 123, 276, 277, 306] [87, 111, 124, 277, 278, 307] [88, 112, 125, 278, 279, 308] [89, 113, 126, 279, 280, 309] [90, 114, 127, 280, 281, 310] [91, 115, 128, 281, 282, 311] [92, 116, 129, 282, 283, 312] [93, 117, 130, 283, 284, 313] [94, 118, 131, 284, 285, 314] [95, 119, 132, 285, 286, 315] [96, 120, 133, 286, 287, 316] [100, 124, 137, 290, 291, 321] [101, 125, 138, 291, 292, 322] [102, 126, 139, 292, 293, 323] [103, 127, 140, 293, 294, 324] [104, 128, 141, 294, 295, 325] [105, 129, 142, 295, 296, 326] [106, 130, 143, 296, 297, 327] [107, 131, 144, 297, 298, 328] [108, 132, 145, 298, 299, 329] [109, 133, 146, 299, 300, 330] [112, 136, 149, 303, 304, 334] [113, 137, 150, 304, 305, 335] [114, 138, 151, 305, 306, 336] [115, 139, 152, 306, 307, 337] [116, 140, 153, 307, 308, 338] [117, 141, 154, 308, 309, 339] [118, 142, 155, 309, 310, 340] [119, 143, 156, 310, 311, 341] [120, 144, 157, 311, 312, 342] [121, 145, 158, 312, 313, 343] [122, 146, 159, 313, 314, 344] [125, 149, 162, 317, 318, 346] [126, 150, 163, 318, 319, 347] [127, 151, 164, 319, 320, 348] [128, 152, 165, 320, 321, 349] [129, 153, 166, 321, 322, 350] [130, 154, 167, 322, 323, 351] [131, 155, 168, 323, 324, 352] [132, 156, 169, 324, 325] [133, 157, 170, 325, 326, 353] [134, 158, 171, 326, 327] [135, 159, 172, 327, 328] [0, 176] [1, 177] [2, 178] [3, 179] [4, 180] [5, 181] [14, 192] [6, 182] [16, 194] [0, 8, 186] [1, 9, 187] [2, 10, 188] [3, 11, 189] [4, 12, 190] [5, 13, 191] [6, 15, 193] [138, 162, 331, 332, 356] [139, 163, 332, 333, 357] [140, 164, 333, 334, 358] [141, 165, 334, 335, 359] [142, 166, 335, 336, 360] [143, 167, 336, 337, 361] [144, 168, 337, 338, 362] [145, 169, 338, 339, 363] [146, 170, 339, 340] [147, 171, 340, 341] [148, 172, 341, 342] [165, 354, 355] [166, 355, 356] [167, 356, 357] [168, 357, 358] [169, 358, 359] [170, 359, 360] [171, 360, 361] [172, 361, 362] [173, 363] [153, 345, 346] [154, 346, 347] [155, 347, 348] [156, 348, 349] [157, 349, 350] [158, 350, 351] [159, 351, 352] [160, 352] [161, 353] [17, 29, 182, 207] [18, 30, 182, 208]
H_Z (179 checks, sparse supports)
[0, 22, 23, 176, 186, 210] [1, 23, 24, 177, 187, 211] [2, 24, 25, 178, 188, 212] [3, 25, 26, 179, 189, 213] [4, 26, 27, 180, 190, 214] [5, 27, 28, 181, 191, 215] [6, 29, 30, 182, 193, 217] [7, 31, 32, 219] [8, 35, 36, 186, 198, 223] [9, 36, 37, 187, 199, 224] [10, 37, 38, 188, 200, 225] [11, 38, 39, 189, 201, 226] [12, 39, 40, 190, 202, 227] [13, 40, 41, 191, 203, 228] [14, 41, 42, 192, 204, 229] [15, 42, 43, 193, 205, 230] [16, 43, 44, 194, 206, 231] [17, 44, 45, 207, 232] [20, 48, 49, 198, 212, 236] [21, 49, 50, 199, 213, 237] [22, 50, 51, 200, 214, 238] [23, 51, 52, 201, 215, 239] [24, 52, 53, 202, 216, 240] [25, 53, 54, 203, 217, 241] [26, 54, 55, 204, 218, 242] [27, 55, 56, 205, 219, 243] [28, 56, 57, 206, 220, 244] [29, 57, 207, 221, 245] [30, 208, 222, 246] [33, 60, 61, 212, 225, 250] [34, 61, 62, 213, 226, 251] [35, 62, 63, 214, 227, 252] [36, 63, 64, 215, 228, 253] [37, 64, 65, 216, 229, 254] [38, 65, 66, 217, 230, 255] [39, 66, 67, 218, 231, 256] [40, 67, 68, 219, 232, 257] [41, 68, 69, 220, 233, 258] [42, 69, 70, 221, 234, 259] [43, 70, 222, 260] [46, 73, 74, 225, 238, 264] [47, 74, 75, 226, 239, 265] [48, 75, 76, 227, 240, 266] [49, 76, 77, 228, 241, 267] [50, 77, 78, 229, 242, 268] [51, 78, 79, 230, 243, 269] [52, 79, 80, 231, 244, 270] [53, 80, 81, 232, 245, 271] [54, 81, 82, 233, 246, 272] [55, 82, 83, 234, 247, 273] [58, 86, 87, 238, 252, 277] [59, 87, 88, 239, 253, 278] [60, 88, 89, 240, 254, 279] [61, 89, 90, 241, 255, 280] [62, 90, 91, 242, 256, 281] [63, 91, 92, 243, 257, 282] [64, 92, 93, 244, 258, 283] [65, 93, 94, 245, 259, 284] [66, 94, 95, 246, 260, 285] [67, 95, 96, 247, 261, 286] [71, 99, 100, 252, 266, 290] [72, 100, 101, 253, 267, 291] [73, 101, 102, 254, 268, 292] [74, 102, 103, 255, 269, 293] [75, 103, 104, 256, 270, 294] [76, 104, 105, 257, 271, 295] [77, 105, 106, 258, 272, 296] [78, 106, 107, 259, 273, 297] [79, 107, 108, 260, 274, 298] [80, 108, 109, 261, 275, 299] [84, 112, 113, 266, 279, 304] [85, 113, 114, 267, 280, 305] [86, 114, 115, 268, 281, 306] [87, 115, 116, 269, 282, 307] [88, 116, 117, 270, 283, 308] [89, 117, 118, 271, 284, 309] [90, 118, 119, 272, 285, 310] [91, 119, 120, 273, 286, 311] [92, 120, 121, 274, 287, 312] [93, 121, 122, 275, 288, 313] [97, 125, 126, 279, 292, 318] [98, 126, 127, 280, 293, 319] [99, 127, 128, 281, 294, 320] [100, 128, 129, 282, 295, 321] [101, 129, 130, 283, 296, 322] [102, 130, 131, 284, 297, 323] [103, 131, 132, 285, 298, 324] [104, 132, 133, 286, 299, 325] [105, 133, 134, 287, 300, 326] [106, 134, 135, 288, 301, 327] [107, 135, 289, 302, 328] [110, 138, 139, 292, 306, 332] [111, 139, 140, 293, 307, 333] [112, 140, 141, 294, 308, 334] [113, 141, 142, 295, 309, 335] [114, 142, 143, 296, 310, 336] [115, 143, 144, 297, 311, 337] [116, 144, 145, 298, 312, 338] [117, 145, 146, 299, 313, 339] [118, 146, 147, 300, 314, 340] [119, 147, 148, 301, 315, 341] [123, 151, 152, 306, 320] [124, 152, 153, 307, 321, 345] [125, 153, 154, 308, 322, 346] [126, 154, 155, 309, 323, 347] [127, 155, 156, 310, 324, 348] [128, 156, 157, 311, 325, 349] [129, 157, 158, 312, 326, 350] [130, 158, 159, 313, 327, 351] [131, 159, 160, 314, 328, 352] [133, 161, 316, 330, 353] [136, 164, 165, 320, 334, 354] [137, 165, 166, 321, 335, 355] [138, 166, 167, 322, 336, 356] [139, 167, 168, 323, 337, 357] [140, 168, 169, 324, 338, 358] [141, 169, 170, 325, 339, 359] [142, 170, 171, 326, 340, 360] [143, 171, 172, 327, 341, 361] [144, 172, 328, 342, 362] [145, 173, 329, 343, 363] [20, 183] [33, 183, 195] [46, 195, 209] [58, 209] [71, 235] [84, 235, 249] [97, 249, 263] [110, 263, 276] [123, 276] [136, 303] [149, 303, 317] [162, 317, 331] [20, 21, 174, 184] [33, 34, 184, 196] [46, 47, 196, 210] [58, 59, 210, 223] [71, 72, 223, 236] [84, 85, 236, 250] [97, 98, 250, 264] [110, 111, 264, 277] [123, 124, 277, 290] [136, 137, 290, 304] [149, 150, 304, 318] [162, 163, 318, 332] [21, 22, 175, 185, 209] [34, 35, 185, 197] [47, 48, 197, 211, 235] [59, 60, 211, 224, 249] [72, 73, 224, 237, 263] [85, 86, 237, 251, 276] [98, 99, 251, 265] [111, 112, 265, 278, 303] [124, 125, 278, 291, 317] [137, 138, 291, 305, 331] [150, 151, 305, 319] [163, 164, 319, 333] [32, 248] [45, 262] [70, 289] [83, 302] [96, 316] [109, 330] [122, 344] [19, 234] [31, 247] [44, 261] [57, 275] [69, 288] [82, 301] [95, 315] [108, 329] [121, 343] [18, 45, 208, 233] [56, 83, 248, 274] [68, 96, 248, 262, 287] [81, 109, 262, 300] [94, 122, 289, 314] [120, 148, 302, 316, 342]
Code ID 364-13-11 · download JSON · raw on GitHub