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[[432,4,23]] d ≤
n
432
k
4
d
23
kd²/n
4.898
w
5
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 23, d_Z ≤ 23 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 23 · witness weight 23 (claimed upper_bound)
witness operator (support, 23 qubits)
[29, 49, 52, 55, 81, 88, 114, 124, 140, 147, 150, 176, 202, 206, 221, 234, 250, 272, 285, 289, 386, 399, 415]
d_Z 23 · witness weight 23 (claimed upper_bound)
witness operator (support, 23 qubits)
[0, 34, 38, 51, 73, 89, 102, 199, 215, 217, 220, 223, 243, 282, 298, 312, 338, 341, 348, 364, 374, 400, 419]
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 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×216 (2,2)×216 (3,2)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 216 (1,3): 216 (2,2): 216 (2,3): 1296 (2,4): 648 (3,2): 216 (3,3): 4104 (3,4): 7128 (3,5): 2592 (3,6): 1296 (3,7): 216
trapping sets H_Z (1,2)×216 (2,2)×216 (3,2)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 216 (1,3): 216 (2,2): 216 (2,3): 1296 (2,4): 648 (3,2): 216 (3,3): 4104 (3,4): 7128 (3,5): 2592 (3,6): 1296 (3,7): 216

Construction & provenance

authors @cbjuan and @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle code over F_2[x,y]/(x18-1, y12-1) with mixed-monomial weight-five polynomials A = 1 + x4 y3, B = 1 + x5 y11 + x10 y, lattice (18,12), taken from Table 3 of arXiv:2610.06623v1. The paper reports this presentation with the interval 5 <= d <= 23 (evidence L4+I+W+B) and no exact certificate. The distance claimed here is an upper bound from this repository's own witness search, not the paper's interval endpoint. Construction by Juan Cruz-Benito (@cbjuan); the independent re-derivation and the distance witnesses are by @MathysRennela.
model human (claimed, not verified)
date 2026-10-06
notes Known parameters: the (n,k,d) triple and the explicit polynomial presentation are published in arXiv:2610.06623v1, Table 3. This entry is an independent reconstruction and re-measurement, not a new construction. The claimed distance is a witness-backed upper bound; the paper states no exact certificate for this row.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (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

[[432,4,23]] — weight-5 bivariate bicycle code from arXiv:2610.06623v1 Table 3

Direction & hypothesis

Target: the unrestricted x weight-6 CSS cell, and inside it the thin k = 4 corner at large n. Screening arXiv:2610.06623v1 (Cruz-Benito, "Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes") showed that every exact code the paper certifies at weight five was either already on this board or Pareto-dominated by it, with one exception: the rows carrying *interval* evidence rather than an exact certificate. Those rows report no exact distance, so nothing on the board claims them, and the paper's own arithmetic (kd^2/n at most 4.90) put them below the cell's headline figures. The expectation was a narrow but real opening at k = 4, n around 420-432, where the board had nothing.

This is a thin-cell record, not a strong one. kd^2/n is 4.90 where the same cell already holds 30.5 ([[672,20,32]]).

What was searched

No fresh search. The direction was to rebuild, verify, and Pareto-screen the 26 explicit presentations printed in Tables 3 and 6 of arXiv:2610.06623v1.

All 10 Table 3 (CSS) presentations were rebuilt with research/kit/bb.py:build_bb and their (n, k, w) measured from the resulting matrices rather than read off the paper; all 10 reproduced their claimed parameters exactly. The replicated-component rows were then dropped, following the paper's own connectivity analysis (409 of its 1,142 catalogue entries are disconnected, and 73.1% of the classes with exact certificates are repeated components).

The 16 Table 6 (PBB) rows were screened on their published triples only. The four-block layout in the paper's Eq. (4) did not reconcile with n = 2*l*m under any standard block arrangement tried here, so no PBB presentation was re-derived and none is claimed. Every one of them is dominated by the board regardless.

Of the surviving CSS rows, seven are dominated by existing entries, two are already on the board, and this one is Pareto-undominated.

Evidence trail

The presentation: l = 18, m = 12, A = 1 + x^4 y^3, `B = 1 + x^5 y^11 + x^10 y, so n = 2*18*12 = 432`.

  • Rebuilt and measured: n = 432, k = 4, max check weight 5, one connected

    • component (no replicated decomposition).

  • Distance: the paper reports 5 <= d <= 23 (its L4+I+W+B evidence) and no

    • exact certificate. This repository's own searches (research/kit/surrogate.py:distance_rand_witness, 4 seeds x 6000 RIS trials, pair_depth=16, gf2_fast backend) independently returned weight-23 logicals on both sides and nothing lighter.

  • Submitted as a witness-backed upper bound, d <= 23, on both sides. It

    • is not certified exact and is not described as such.

  • The trusted gate (verify/validate_candidate.py) returns passed: true,

    • with dedup clean, refutation unrefuted over 8000 RIS trials, and novelty.board_advancing: true on advances_by: ["d", "n"].

Dead end worth recording: the paper's other interval row, [[420,4,5<=d<=22]] (two distinct presentations of it), shares this row's cell and is filed as a separate [[420,4,22]] entry. The exact [[432,4,10]] presentation from the same table is dominated eleven times over by entries such as [[192,4,16]].

Tools

Human-directed agent work (opencode CLI) rebuilding and screening published constructions; research/kit/bb.py for construction, research/kit/surrogate.py and the gf2_fast accelerator for the witness search, verify/validate_candidate.py for the gate. Roughly 25 CPU-minutes of witness search on an n=432 code.

Reproduction

import sys; sys.path[:0] = ["research/kit"]
from bb import build_bb
HX, HZ = build_bb(18, 12, [(4, 3), (0, 0)], [(5, 11), (10, 1), (0, 0)])

Terms are (x, y) exponent pairs against F_2[x,y]/(x^18 - 1, y^12 - 1); H_X = [A|B], H_Z = [B^T|A^T] per research/kit/bb.py. Source of the presentation: arXiv:2610.06623v1, Table 3. Construction by Juan Cruz-Benito (@cbjuan); the reconstruction and the distance witnesses are by @MathysRennela.

Parity checks

X-checks 216 (max weight 5) · Z-checks 216 (max weight 5)
H_X (216 checks, sparse supports)
[0, 51, 216, 287, 337] [1, 52, 217, 276, 338] [2, 53, 218, 277, 339] [3, 54, 219, 278, 340] [4, 55, 220, 279, 341] [5, 56, 221, 280, 342] [6, 57, 222, 281, 343] [7, 58, 223, 282, 344] [8, 59, 224, 283, 345] [9, 48, 225, 284, 346] [10, 49, 226, 285, 347] [11, 50, 227, 286, 336] [12, 63, 228, 299, 349] [13, 64, 229, 288, 350] [14, 65, 230, 289, 351] [15, 66, 231, 290, 352] [16, 67, 232, 291, 353] [17, 68, 233, 292, 354] [18, 69, 234, 293, 355] [19, 70, 235, 294, 356] [20, 71, 236, 295, 357] [21, 60, 237, 296, 358] [22, 61, 238, 297, 359] [23, 62, 239, 298, 348] [24, 75, 240, 311, 361] [25, 76, 241, 300, 362] [26, 77, 242, 301, 363] [27, 78, 243, 302, 364] [28, 79, 244, 303, 365] [29, 80, 245, 304, 366] [30, 81, 246, 305, 367] [31, 82, 247, 306, 368] [32, 83, 248, 307, 369] [33, 72, 249, 308, 370] [34, 73, 250, 309, 371] [35, 74, 251, 310, 360] [36, 87, 252, 323, 373] [37, 88, 253, 312, 374] [38, 89, 254, 313, 375] [39, 90, 255, 314, 376] [40, 91, 256, 315, 377] [41, 92, 257, 316, 378] [42, 93, 258, 317, 379] [43, 94, 259, 318, 380] [44, 95, 260, 319, 381] [45, 84, 261, 320, 382] [46, 85, 262, 321, 383] [47, 86, 263, 322, 372] [48, 99, 264, 335, 385] [49, 100, 265, 324, 386] [50, 101, 266, 325, 387] [51, 102, 267, 326, 388] [52, 103, 268, 327, 389] [53, 104, 269, 328, 390] [54, 105, 270, 329, 391] [55, 106, 271, 330, 392] [56, 107, 272, 331, 393] [57, 96, 273, 332, 394] [58, 97, 274, 333, 395] [59, 98, 275, 334, 384] [60, 111, 276, 347, 397] [61, 112, 277, 336, 398] [62, 113, 278, 337, 399] [63, 114, 279, 338, 400] [64, 115, 280, 339, 401] [65, 116, 281, 340, 402] [66, 117, 282, 341, 403] [67, 118, 283, 342, 404] [68, 119, 284, 343, 405] [69, 108, 285, 344, 406] [70, 109, 286, 345, 407] [71, 110, 287, 346, 396] [72, 123, 288, 359, 409] [73, 124, 289, 348, 410] [74, 125, 290, 349, 411] [75, 126, 291, 350, 412] [76, 127, 292, 351, 413] [77, 128, 293, 352, 414] [78, 129, 294, 353, 415] [79, 130, 295, 354, 416] [80, 131, 296, 355, 417] [81, 120, 297, 356, 418] [82, 121, 298, 357, 419] [83, 122, 299, 358, 408] [84, 135, 300, 371, 421] [85, 136, 301, 360, 422] [86, 137, 302, 361, 423] [87, 138, 303, 362, 424] [88, 139, 304, 363, 425] [89, 140, 305, 364, 426] [90, 141, 306, 365, 427] [91, 142, 307, 366, 428] [92, 143, 308, 367, 429] [93, 132, 309, 368, 430] [94, 133, 310, 369, 431] [95, 134, 311, 370, 420] [96, 147, 217, 312, 383] [97, 148, 218, 313, 372] [98, 149, 219, 314, 373] [99, 150, 220, 315, 374] [100, 151, 221, 316, 375] [101, 152, 222, 317, 376] [102, 153, 223, 318, 377] [103, 154, 224, 319, 378] [104, 155, 225, 320, 379] [105, 144, 226, 321, 380] [106, 145, 227, 322, 381] [107, 146, 216, 323, 382] [108, 159, 229, 324, 395] [109, 160, 230, 325, 384] [110, 161, 231, 326, 385] [111, 162, 232, 327, 386] [112, 163, 233, 328, 387] [113, 164, 234, 329, 388] [114, 165, 235, 330, 389] [115, 166, 236, 331, 390] [116, 167, 237, 332, 391] [117, 156, 238, 333, 392] [118, 157, 239, 334, 393] [119, 158, 228, 335, 394] [120, 171, 241, 336, 407] [121, 172, 242, 337, 396] [122, 173, 243, 338, 397] [123, 174, 244, 339, 398] [124, 175, 245, 340, 399] [125, 176, 246, 341, 400] [126, 177, 247, 342, 401] [127, 178, 248, 343, 402] [128, 179, 249, 344, 403] [129, 168, 250, 345, 404] [130, 169, 251, 346, 405] [131, 170, 240, 347, 406] [132, 183, 253, 348, 419] [133, 184, 254, 349, 408] [134, 185, 255, 350, 409] [135, 186, 256, 351, 410] [136, 187, 257, 352, 411] [137, 188, 258, 353, 412] [138, 189, 259, 354, 413] [139, 190, 260, 355, 414] [140, 191, 261, 356, 415] [141, 180, 262, 357, 416] [142, 181, 263, 358, 417] [143, 182, 252, 359, 418] [144, 195, 265, 360, 431] [145, 196, 266, 361, 420] [146, 197, 267, 362, 421] [147, 198, 268, 363, 422] [148, 199, 269, 364, 423] [149, 200, 270, 365, 424] [150, 201, 271, 366, 425] [151, 202, 272, 367, 426] [152, 203, 273, 368, 427] [153, 192, 274, 369, 428] [154, 193, 275, 370, 429] [155, 194, 264, 371, 430] [156, 207, 227, 277, 372] [157, 208, 216, 278, 373] [158, 209, 217, 279, 374] [159, 210, 218, 280, 375] [160, 211, 219, 281, 376] [161, 212, 220, 282, 377] [162, 213, 221, 283, 378] [163, 214, 222, 284, 379] [164, 215, 223, 285, 380] [165, 204, 224, 286, 381] [166, 205, 225, 287, 382] [167, 206, 226, 276, 383] [3, 168, 239, 289, 384] [4, 169, 228, 290, 385] [5, 170, 229, 291, 386] [6, 171, 230, 292, 387] [7, 172, 231, 293, 388] [8, 173, 232, 294, 389] [9, 174, 233, 295, 390] [10, 175, 234, 296, 391] [11, 176, 235, 297, 392] [0, 177, 236, 298, 393] [1, 178, 237, 299, 394] [2, 179, 238, 288, 395] [15, 180, 251, 301, 396] [16, 181, 240, 302, 397] [17, 182, 241, 303, 398] [18, 183, 242, 304, 399] [19, 184, 243, 305, 400] [20, 185, 244, 306, 401] [21, 186, 245, 307, 402] [22, 187, 246, 308, 403] [23, 188, 247, 309, 404] [12, 189, 248, 310, 405] [13, 190, 249, 311, 406] [14, 191, 250, 300, 407] [27, 192, 263, 313, 408] [28, 193, 252, 314, 409] [29, 194, 253, 315, 410] [30, 195, 254, 316, 411] [31, 196, 255, 317, 412] [32, 197, 256, 318, 413] [33, 198, 257, 319, 414] [34, 199, 258, 320, 415] [35, 200, 259, 321, 416] [24, 201, 260, 322, 417] [25, 202, 261, 323, 418] [26, 203, 262, 312, 419] [39, 204, 275, 325, 420] [40, 205, 264, 326, 421] [41, 206, 265, 327, 422] [42, 207, 266, 328, 423] [43, 208, 267, 329, 424] [44, 209, 268, 330, 425] [45, 210, 269, 331, 426] [46, 211, 270, 332, 427] [47, 212, 271, 333, 428] [36, 213, 272, 334, 429] [37, 214, 273, 335, 430] [38, 215, 274, 324, 431]
H_Z (216 checks, sparse supports)
[0, 107, 157, 216, 393] [1, 96, 158, 217, 394] [2, 97, 159, 218, 395] [3, 98, 160, 219, 384] [4, 99, 161, 220, 385] [5, 100, 162, 221, 386] [6, 101, 163, 222, 387] [7, 102, 164, 223, 388] [8, 103, 165, 224, 389] [9, 104, 166, 225, 390] [10, 105, 167, 226, 391] [11, 106, 156, 227, 392] [12, 119, 169, 228, 405] [13, 108, 170, 229, 406] [14, 109, 171, 230, 407] [15, 110, 172, 231, 396] [16, 111, 173, 232, 397] [17, 112, 174, 233, 398] [18, 113, 175, 234, 399] [19, 114, 176, 235, 400] [20, 115, 177, 236, 401] [21, 116, 178, 237, 402] [22, 117, 179, 238, 403] [23, 118, 168, 239, 404] [24, 131, 181, 240, 417] [25, 120, 182, 241, 418] [26, 121, 183, 242, 419] [27, 122, 184, 243, 408] [28, 123, 185, 244, 409] [29, 124, 186, 245, 410] [30, 125, 187, 246, 411] [31, 126, 188, 247, 412] [32, 127, 189, 248, 413] [33, 128, 190, 249, 414] [34, 129, 191, 250, 415] [35, 130, 180, 251, 416] [36, 143, 193, 252, 429] [37, 132, 194, 253, 430] [38, 133, 195, 254, 431] [39, 134, 196, 255, 420] [40, 135, 197, 256, 421] [41, 136, 198, 257, 422] [42, 137, 199, 258, 423] [43, 138, 200, 259, 424] [44, 139, 201, 260, 425] [45, 140, 202, 261, 426] [46, 141, 203, 262, 427] [47, 142, 192, 263, 428] [48, 155, 205, 225, 264] [49, 144, 206, 226, 265] [50, 145, 207, 227, 266] [51, 146, 208, 216, 267] [52, 147, 209, 217, 268] [53, 148, 210, 218, 269] [54, 149, 211, 219, 270] [55, 150, 212, 220, 271] [56, 151, 213, 221, 272] [57, 152, 214, 222, 273] [58, 153, 215, 223, 274] [59, 154, 204, 224, 275] [1, 60, 167, 237, 276] [2, 61, 156, 238, 277] [3, 62, 157, 239, 278] [4, 63, 158, 228, 279] [5, 64, 159, 229, 280] [6, 65, 160, 230, 281] [7, 66, 161, 231, 282] [8, 67, 162, 232, 283] [9, 68, 163, 233, 284] [10, 69, 164, 234, 285] [11, 70, 165, 235, 286] [0, 71, 166, 236, 287] [13, 72, 179, 249, 288] [14, 73, 168, 250, 289] [15, 74, 169, 251, 290] [16, 75, 170, 240, 291] [17, 76, 171, 241, 292] [18, 77, 172, 242, 293] [19, 78, 173, 243, 294] [20, 79, 174, 244, 295] [21, 80, 175, 245, 296] [22, 81, 176, 246, 297] [23, 82, 177, 247, 298] [12, 83, 178, 248, 299] [25, 84, 191, 261, 300] [26, 85, 180, 262, 301] [27, 86, 181, 263, 302] [28, 87, 182, 252, 303] [29, 88, 183, 253, 304] [30, 89, 184, 254, 305] [31, 90, 185, 255, 306] [32, 91, 186, 256, 307] [33, 92, 187, 257, 308] [34, 93, 188, 258, 309] [35, 94, 189, 259, 310] [24, 95, 190, 260, 311] [37, 96, 203, 273, 312] [38, 97, 192, 274, 313] [39, 98, 193, 275, 314] [40, 99, 194, 264, 315] [41, 100, 195, 265, 316] [42, 101, 196, 266, 317] [43, 102, 197, 267, 318] [44, 103, 198, 268, 319] [45, 104, 199, 269, 320] [46, 105, 200, 270, 321] [47, 106, 201, 271, 322] [36, 107, 202, 272, 323] [49, 108, 215, 285, 324] [50, 109, 204, 286, 325] [51, 110, 205, 287, 326] [52, 111, 206, 276, 327] [53, 112, 207, 277, 328] [54, 113, 208, 278, 329] [55, 114, 209, 279, 330] [56, 115, 210, 280, 331] [57, 116, 211, 281, 332] [58, 117, 212, 282, 333] [59, 118, 213, 283, 334] [48, 119, 214, 284, 335] [11, 61, 120, 297, 336] [0, 62, 121, 298, 337] [1, 63, 122, 299, 338] [2, 64, 123, 288, 339] [3, 65, 124, 289, 340] [4, 66, 125, 290, 341] [5, 67, 126, 291, 342] [6, 68, 127, 292, 343] [7, 69, 128, 293, 344] [8, 70, 129, 294, 345] [9, 71, 130, 295, 346] [10, 60, 131, 296, 347] [23, 73, 132, 309, 348] [12, 74, 133, 310, 349] [13, 75, 134, 311, 350] [14, 76, 135, 300, 351] [15, 77, 136, 301, 352] [16, 78, 137, 302, 353] [17, 79, 138, 303, 354] [18, 80, 139, 304, 355] [19, 81, 140, 305, 356] [20, 82, 141, 306, 357] [21, 83, 142, 307, 358] [22, 72, 143, 308, 359] [35, 85, 144, 321, 360] [24, 86, 145, 322, 361] [25, 87, 146, 323, 362] [26, 88, 147, 312, 363] [27, 89, 148, 313, 364] [28, 90, 149, 314, 365] [29, 91, 150, 315, 366] [30, 92, 151, 316, 367] [31, 93, 152, 317, 368] [32, 94, 153, 318, 369] [33, 95, 154, 319, 370] [34, 84, 155, 320, 371] [47, 97, 156, 333, 372] [36, 98, 157, 334, 373] [37, 99, 158, 335, 374] [38, 100, 159, 324, 375] [39, 101, 160, 325, 376] [40, 102, 161, 326, 377] [41, 103, 162, 327, 378] [42, 104, 163, 328, 379] [43, 105, 164, 329, 380] [44, 106, 165, 330, 381] [45, 107, 166, 331, 382] [46, 96, 167, 332, 383] [59, 109, 168, 345, 384] [48, 110, 169, 346, 385] [49, 111, 170, 347, 386] [50, 112, 171, 336, 387] [51, 113, 172, 337, 388] [52, 114, 173, 338, 389] [53, 115, 174, 339, 390] [54, 116, 175, 340, 391] [55, 117, 176, 341, 392] [56, 118, 177, 342, 393] [57, 119, 178, 343, 394] [58, 108, 179, 344, 395] [71, 121, 180, 357, 396] [60, 122, 181, 358, 397] [61, 123, 182, 359, 398] [62, 124, 183, 348, 399] [63, 125, 184, 349, 400] [64, 126, 185, 350, 401] [65, 127, 186, 351, 402] [66, 128, 187, 352, 403] [67, 129, 188, 353, 404] [68, 130, 189, 354, 405] [69, 131, 190, 355, 406] [70, 120, 191, 356, 407] [83, 133, 192, 369, 408] [72, 134, 193, 370, 409] [73, 135, 194, 371, 410] [74, 136, 195, 360, 411] [75, 137, 196, 361, 412] [76, 138, 197, 362, 413] [77, 139, 198, 363, 414] [78, 140, 199, 364, 415] [79, 141, 200, 365, 416] [80, 142, 201, 366, 417] [81, 143, 202, 367, 418] [82, 132, 203, 368, 419] [95, 145, 204, 381, 420] [84, 146, 205, 382, 421] [85, 147, 206, 383, 422] [86, 148, 207, 372, 423] [87, 149, 208, 373, 424] [88, 150, 209, 374, 425] [89, 151, 210, 375, 426] [90, 152, 211, 376, 427] [91, 153, 212, 377, 428] [92, 154, 213, 378, 429] [93, 155, 214, 379, 430] [94, 144, 215, 380, 431]
Code ID 432-4-23 · download JSON · raw on GitHub