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[[420,6,16]] d ≤
n
420
k
6
d
16
kd²/n
3.657
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[18, 22, 26, 34, 214, 216, 230, 277, 279, 293, 326, 328, 342, 389, 391, 405]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[7, 9, 42, 56, 58, 105, 119, 121, 140, 154, 156, 203, 301, 309, 313, 317]
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)×210 (2,2)×210 (3,2)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 210 (1,3): 210 (2,2): 210 (2,3): 1260 (2,4): 630 (3,2): 210 (3,3): 3990 (3,4): 6930 (3,5): 2520 (3,6): 1260 (3,7): 210
trapping sets H_Z (1,2)×210 (2,2)×210 (3,2)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 210 (1,3): 210 (2,2): 210 (2,3): 1260 (2,4): 630 (3,2): 210 (3,3): 3990 (3,4): 6930 (3,5): 2520 (3,6): 1260 (3,7): 210

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle code over F_2[x,y]/(x15-1, y14-1) with mixed-monomial weight-five polynomials A = 1 + x4 y7, B = 1 + x14 y2 + x14 y6, lattice (15,14), taken from Table 3 of arXiv:2610.06623v1. The paper reports this presentation with the interval 5 <= d <= 16 (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-dcca76b0. 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, with no independent witness search performed; this repository's own prior experience is that such bounds can collapse under scrutiny (9 of 10 audited B-tier bounds in the same campaign did). 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-16 logical operator on each side, confirming the paper's upper bound holds rather than collapsing. k was independently re-derived as rank(H_X)+rank(H_Z) subtracted from n, matching the paper's k=6.
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

[[420,6,16]] — 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]] and [[420,4,22]] entries (merged 2026-10-06) already showed that several connected, non-dominated rows in that catalogue carry only *interval* distance evidence (a MILP incumbent or a raw decoder scalar, not an independently reconstructed witness) rather than an exact certificate, so nothing on the board claims them yet. This row, the paper's class CL-dcca76b0, is one of those: k = 6 at n = 420, with the paper's own catalogue recording only an MILP-incumbent upper bound and no independent stress-testing on record for this specific row.

What was searched

No fresh search for a new construction. Direct re-verification of one already-published presentation: l = 15, m = 14, A = 1 + x^4 y^7, B = 1 + x^14 y^2 + x^14 y^6, so n = 2*15*14 = 420.

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 = 6 exactly.

Evidence trail

  • Rebuilt and measured: n = 420, k = 6, max check weight 5, one

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

  • The paper's own catalogue records 5 <= d <= 16 for this row: lower

    • bound from an exhaustive weight-four exclusion, upper bound from an unverified MILP incumbent with no persisted logical operator and, as far as the paper's own provenance records show, no independent search run against it before this submission.

  • 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-16 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 16 across the full search. The MILP incumbent's upper bound holds; it did not collapse.

  • Submitted as a witness-backed upper bound, d <= 16, on both sides.

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

  • uv run python verify/qldpc_verify.py codes/420-6-16.json passes locally

    • with ok: true, earned_distance.d = {value: 16, 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-16 logical on the first full pass and nothing lighter. A sibling row at a different point, [[420,8,14]], surfaced in the same screening pass and is intentionally not submitted alongside this one: its only distance evidence is a raw decoder scalar with no witness at all, so it needs the same hardening before it is trustworthy enough to submit.

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=420 code.

Reproduction

import sys; sys.path[:0] = ["research/kit"]
from bb import build_bb
HX, HZ = build_bb(15, 14, [(4, 7), (0, 0)], [(14, 2), (14, 6), (0, 0)])

Terms are (x, y) exponent pairs against F_2[x,y]/(x^15 - 1, y^14 - 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-dcca76b0. Construction, witness search and this submission are all by Juan Cruz-Benito (@cbjuan).

Parity checks

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