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[[432,8,15]] d ≤
n
432
k
8
d
15
kd²/n
4.167
w
5
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[43, 48, 65, 82, 101, 123, 237, 259, 279, 301, 303, 333, 355, 357, 399]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[8, 28, 50, 104, 146, 148, 170, 200, 202, 222, 239, 244, 380, 402, 421]
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
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 y6, B = 1 + x y5 + x5 y10, lattice (18,12), taken from Table 3 of arXiv:2610.06623v1. The paper reports this presentation with the interval 5 <= d <= 15 (evidence L4+I) 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 and distance witnesses by Juan Cruz-Benito (@cbjuan).
model human (claimed, not verified)
builds on arXiv:2610.06623v1
date 2026-10-07
notes Known parameters: the (n,k,d) triple and the explicit polynomial presentation are published in arXiv:2610.06623v1, Table 3, class CL-ab07a1d1. This entry is an independent reconstruction and re-measurement, not a new construction. The paper's own catalogue carried only an unverified MILP-incumbent (I-tier) upper bound for this row. An independent randomized GF(2) kernel/permutation witness search (8,000 trials/side) rebuilt the check matrices from the published polynomials and found a weight-15 logical operator on each side, confirming the paper's upper bound holds. k was independently re-derived from the rebuilt check matrices, matching the paper's k=8.
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

[[432,8,15]] — weight-5 bivariate bicycle code from arXiv:2610.06623v1 Table 3

Direction & hypothesis

Target: the unrestricted x weight-6 CSS cell. arXiv:2610.06623v1 (Cruz-Benito, "Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes") catalogues 1,142 weight-five BB presentations in 630 colored-BLISS classes; this repository's own [[432,4,23]], [[420,4,22]] and [[420,6,16]] entries already showed that several connected, non-dominated rows in that catalogue carry only *interval* distance evidence (a MILP incumbent, not an independently reconstructed witness) rather than an exact certificate. This row, the paper's class CL-ab07a1d1, is another: k = 8 at n = 432, with the paper's own catalogue recording only an unverified MILP-incumbent upper bound and no persisted logical operator.

What was searched

No fresh search for a new construction. Direct re-verification of one already-published presentation: l = 18, m = 12, A = 1 + x^4 y^6, B = 1 + x y^5 + x^5 y^10, so n = 2*18*12 = 432.

The check matrices were rebuilt from these polynomials with research/kit/bb.py:build_bb and k was independently re-derived as n - rank(H_X) - rank(H_Z) over GF(2), reproducing the paper's k = 8 exactly.

Evidence trail

  • Rebuilt and measured: n = 432, k = 8, max check weight 5, one

    • connected component (no replicated decomposition — confirmed structurally on the rebuilt matrices).

  • The paper's own catalogue records 5 <= d <= 15 for this row: lower

    • bound from exhaustive weight-four exclusion, upper bound from an unverified MILP incumbent with no persisted witness.

  • An independent randomized search over the kernel of the opposite check

    • matrix (random column permutations each trial, keeping the lightest rows and their pairwise GF(2) combinations under a fresh RREF, 8,000 trials per side, two fixed seeds) found a weight-15 logical operator on each side: zero syndrome against the opposite check matrix, and outside the stabilizer row space, independently re-confirmed by a second, separately coded exclusion check against the same rebuilt matrices. Neither side produced anything lighter than 15. The MILP incumbent's upper bound holds; it did not collapse.

  • Submitted as a witness-backed upper bound, d <= 15, on both sides.

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

  • uv run python verify/qldpc_verify.py codes/432-8-15.json passes locally

    • with ok: true, earned_distance.d = {value: 15, tier: upper_bound}, and one connected css_equivalent image. A full-board verify/verify_all.py pass found no fingerprint or signature collision with any existing entry.

Dead ends

None for this row specifically — the search found a weight-15 logical on the first full pass and nothing lighter.

Tools

Human-directed agent work (Claude Code, Claude Opus/Sonnet 5) rebuilding and witness-hardening a published construction, using research/kit/bb.py for construction and verify/qldpc_verify.py for the local gate. The witness search (the randomized kernel/permutation method described above, 8,000 trials/side, two fixed seeds) was run against the same matrices this PR's research/kit/bb.py call reproduces; the method is specified in full above so it can be rewritten directly against those matrices. Roughly 2 CPU-minutes of 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, 6), (0, 0)], [(1, 5), (5, 10), (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, class CL-ab07a1d1. Construction, witness search and this submission are all by Juan Cruz-Benito (@cbjuan).

Parity checks

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