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[[520,44,8]] d ≤
n
520
k
44
d
8
kd²/n
5.415
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[9, 24, 45, 56, 57, 58, 87, 100]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[91, 99, 112, 118, 205, 216, 320, 341]
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 8 · H_Z 8 (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 2.308) · H_Z 2–3 (mean 2.308)
trapping sets H_X (1,2)×360 (2,2)×720 (3,2)×1440 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 360 (1,3): 160 (2,2): 720 (2,3): 1440 (2,4): 240 (3,2): 1440 (3,3): 7200 (3,4): 4560 (3,5): 1920 (3,6): 720
trapping sets H_Z (1,2)×360 (2,2)×720 (3,2)×1440 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 360 (1,3): 160 (2,2): 720 (2,3): 1440 (2,4): 240 (3,2): 1440 (3,3): 7200 (3,4): 4560 (3,5): 1920 (3,6): 720

Construction & provenance

authors @msilve160
provenance submitted through the challenge
novelty novelty not audited
construction Lifted product of two 2x3 monomial base matrices A, B over F_2[G], G = Z_5 x|_4 Z_8 (metacyclic, relation y x = x4 y) (|G|=40), entries acting by the left (A) and right (B) regular representation (research/kit/nonabelian_lp.py, lifted_product_base). n = 13|G|, check weight 5 (row weight 3 of A + row weight 2 of B, and vice versa). Same construction family as arXiv:2607.28795 (mitten codes) / arXiv:2607.27644 (ZSZ lifted products), at lower entry weight (monomial, not trinomial) than the published weight-9 form. Reproduces a candidate flagged but not submitted in fieldnotes/2026-09-16-lifted-product-girth-cap.md (Finding 5), found via the merged research/kit/nonabelian_lp.sample_nonabelian_lp sampler. This gap in the check-weight-5 region of the board was found by @vprusso (using Claude Fable 5.1), who also wrote research/kit/nonabelian_lp.py and submitted the neighboring codes/624-52-9.json from the same campaign; see fieldnotes/2026-09-16-lifted-product-girth-cap.md for the original search (this submission reproduces one of that campaign's flagged-but-unsubmitted points using the already-merged tooling).
model Claude Claude Sonnet 5 (claimed, not verified)
builds on arXiv:2607.28795, arXiv:2607.27644
date 2026-09-19
family lifted product (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

[[520,44,8]] — non-abelian lifted product over ZSZ(5,8,4)

Credit

This is not an original find: the gap was identified, and the tooling to fill it was written, by @vprusso (using Claude Fable 5.1), in the campaign documented at fieldnotes/2026-09-16-lifted-product-girth-cap.md. @vprusso also wrote research/kit/nonabelian_lp.py (the lifted-product constructor and sampler this submission reuses unmodified) and submitted the neighboring codes/624-52-9.json from the same sweep. That fieldnote explicitly states this exact point was found but left unsubmitted ("left for a later run"); this submission only reruns that already-merged tooling to confirm and package it. No new construction or search method is claimed here.

Direction & hypothesis

Same direction as 416-36-8.note.md: reproducing a candidate explicitly flagged as found-but-unsubmitted in fieldnotes/2026-09-16-lifted-product-girth-cap.md (Finding 5): "the weight-5 points [[416,36,8]], [[520,44,8]] and [[546,46,8]] ... left for a later run." Same 2x3-monomial-base lifted-product construction, check weight 5, rate 1/13.

What was searched

Same sweep as 416-36-8.note.md (1,500 random codes, 76 groups with 31 <= |G| <= 53, A_w = B_w = [[1,1,1],[1,1,1]], staged screening at 400 fast RIS trials). This point was reproduced under several different group presentations at |G|=40 (ZSZ(5,8,4), ZSZ(5,8,2) x2, ZSZ(5,8,3) x2, C5xD4) — the presentation used here is ZSZ(5,8,4).

Evidence trail

  • Screening: d<=8 at 400 fast RIS trials.
  • Deep confirmation: d<=8 held at 500,000 fast RIS trials (seed 1234).
  • Packaging: submit.make_submission witness search (3,000 trials/side,

    • numpy) independently confirmed weight-8 witnesses on both sides.

  • Gate: verify/validate_candidate.py returned passed: true,

    • refute.refuted: false, novelty.board_advancing: true for (weight-6, unrestricted), dominated_by: [].

  • Claim: witness-backed upper bound, not exact.

Dead ends

See 416-36-8.note.md and the fieldnote itself — same sweep, same prior negative results on adjacent profiles.

Tools

Same as 416-36-8.note.md: Claude Sonnet 5, single CPU core, gf2_fast backend, research/kit/nonabelian_lp.py + search.py + submit.py + verify/validate_candidate.py. No new constructor code written.

Reproduction

Group ZSZ(5,8,4): relation y x = x^4 y, x^5 = y^8 = 1, element x^a y^b at index 8a+b, |G|=40.

import sys
sys.path.insert(0, "research/kit")
from nonabelian_lp import zsz, lifted_product_base
from submit import make_submission, save_submission

mul = zsz(5, 8, 4)
A = [[[20], [15], [10]], [[2], [16], [13]]]
B = [[[34], [1], [26]], [[36], [4], [2]]]
HX, HZ = lifted_product_base(mul, A, B)   # n=520, k=44

doc = make_submission(
    HX, HZ,
    name="[[520,44,8]] non-abelian lifted product, ZSZ(5,8,4)",
    construction="Lifted product of two 2x3 monomial base matrices A, B over "
                 "F_2[G], G = Z_5 x|_4 Z_8 (metacyclic), entries acting by the "
                 "left (A) and right (B) regular representation.",
    authors=["@msilve160"], family="lifted-product",
    references=["arXiv:2607.28795", "arXiv:2607.27644"],
    confidence="upper_bound", trials=20000, seed=0,
)
save_submission(doc, "codes/520-44-8.json")   # only after human review + PR

Parity checks

X-checks 240 (max weight 5) · Z-checks 240 (max weight 5)
H_X (240 checks, sparse supports)
[28, 129, 278, 394, 436] [29, 130, 279, 371, 413] [30, 131, 272, 396, 438] [31, 132, 273, 373, 415] [24, 133, 274, 398, 432] [25, 134, 275, 375, 409] [26, 135, 276, 392, 434] [27, 128, 277, 369, 411] [36, 121, 246, 362, 404] [37, 122, 247, 379, 421] [38, 123, 240, 364, 406] [39, 124, 241, 381, 423] [32, 125, 242, 366, 400] [33, 126, 243, 383, 417] [34, 127, 244, 360, 402] [35, 120, 245, 377, 419] [4, 153, 254, 370, 412] [5, 154, 255, 387, 429] [6, 155, 248, 372, 414] [7, 156, 249, 389, 431] [0, 157, 250, 374, 408] [1, 158, 251, 391, 425] [2, 159, 252, 368, 410] [3, 152, 253, 385, 427] [12, 145, 262, 378, 420] [13, 146, 263, 395, 437] [14, 147, 256, 380, 422] [15, 148, 257, 397, 439] [8, 149, 258, 382, 416] [9, 150, 259, 399, 433] [10, 151, 260, 376, 418] [11, 144, 261, 393, 435] [20, 137, 270, 386, 428] [21, 138, 271, 363, 405] [22, 139, 264, 388, 430] [23, 140, 265, 365, 407] [16, 141, 266, 390, 424] [17, 142, 267, 367, 401] [18, 143, 268, 384, 426] [19, 136, 269, 361, 403] [68, 169, 318, 361, 404] [69, 170, 319, 362, 405] [70, 171, 312, 363, 406] [71, 172, 313, 364, 407] [64, 173, 314, 365, 400] [65, 174, 315, 366, 401] [66, 175, 316, 367, 402] [67, 168, 317, 360, 403] [76, 161, 286, 369, 412] [77, 162, 287, 370, 413] [78, 163, 280, 371, 414] [79, 164, 281, 372, 415] [72, 165, 282, 373, 408] [73, 166, 283, 374, 409] [74, 167, 284, 375, 410] [75, 160, 285, 368, 411] [44, 193, 294, 377, 420] [45, 194, 295, 378, 421] [46, 195, 288, 379, 422] [47, 196, 289, 380, 423] [40, 197, 290, 381, 416] [41, 198, 291, 382, 417] [42, 199, 292, 383, 418] [43, 192, 293, 376, 419] [52, 185, 302, 385, 428] [53, 186, 303, 386, 429] [54, 187, 296, 387, 430] [55, 188, 297, 388, 431] [48, 189, 298, 389, 424] [49, 190, 299, 390, 425] [50, 191, 300, 391, 426] [51, 184, 301, 384, 427] [60, 177, 310, 393, 436] [61, 178, 311, 394, 437] [62, 179, 304, 395, 438] [63, 180, 305, 396, 439] [56, 181, 306, 397, 432] [57, 182, 307, 398, 433] [58, 183, 308, 399, 434] [59, 176, 309, 392, 435] [108, 209, 358, 386, 402] [109, 210, 359, 379, 403] [110, 211, 352, 388, 404] [111, 212, 353, 381, 405] [104, 213, 354, 390, 406] [105, 214, 355, 383, 407] [106, 215, 356, 384, 400] [107, 208, 357, 377, 401] [116, 201, 326, 394, 410] [117, 202, 327, 387, 411] [118, 203, 320, 396, 412] [119, 204, 321, 389, 413] [112, 205, 322, 398, 414] [113, 206, 323, 391, 415] [114, 207, 324, 392, 408] [115, 200, 325, 385, 409] [84, 233, 334, 362, 418] [85, 234, 335, 395, 419] [86, 235, 328, 364, 420] [87, 236, 329, 397, 421] [80, 237, 330, 366, 422] [81, 238, 331, 399, 423] [82, 239, 332, 360, 416] [83, 232, 333, 393, 417] [92, 225, 342, 370, 426] [93, 226, 343, 363, 427] [94, 227, 336, 372, 428] [95, 228, 337, 365, 429] [88, 229, 338, 374, 430] [89, 230, 339, 367, 431] [90, 231, 340, 368, 424] [91, 224, 341, 361, 425] [100, 217, 350, 378, 434] [101, 218, 351, 371, 435] [102, 219, 344, 380, 436] [103, 220, 345, 373, 437] [96, 221, 346, 382, 438] [97, 222, 347, 375, 439] [98, 223, 348, 376, 432] [99, 216, 349, 369, 433] [6, 144, 251, 474, 516] [7, 145, 252, 451, 493] [0, 146, 253, 476, 518] [1, 147, 254, 453, 495] [2, 148, 255, 478, 512] [3, 149, 248, 455, 489] [4, 150, 249, 472, 514] [5, 151, 250, 449, 491] [14, 152, 243, 442, 484] [15, 153, 244, 459, 501] [8, 154, 245, 444, 486] [9, 155, 246, 461, 503] [10, 156, 247, 446, 480] [11, 157, 240, 463, 497] [12, 158, 241, 440, 482] [13, 159, 242, 457, 499] [22, 120, 275, 450, 492] [23, 121, 276, 467, 509] [16, 122, 277, 452, 494] [17, 123, 278, 469, 511] [18, 124, 279, 454, 488] [19, 125, 272, 471, 505] [20, 126, 273, 448, 490] [21, 127, 274, 465, 507] [30, 128, 267, 458, 500] [31, 129, 268, 475, 517] [24, 130, 269, 460, 502] [25, 131, 270, 477, 519] [26, 132, 271, 462, 496] [27, 133, 264, 479, 513] [28, 134, 265, 456, 498] [29, 135, 266, 473, 515] [38, 136, 259, 466, 508] [39, 137, 260, 443, 485] [32, 138, 261, 468, 510] [33, 139, 262, 445, 487] [34, 140, 263, 470, 504] [35, 141, 256, 447, 481] [36, 142, 257, 464, 506] [37, 143, 258, 441, 483] [46, 184, 291, 441, 484] [47, 185, 292, 442, 485] [40, 186, 293, 443, 486] [41, 187, 294, 444, 487] [42, 188, 295, 445, 480] [43, 189, 288, 446, 481] [44, 190, 289, 447, 482] [45, 191, 290, 440, 483] [54, 192, 283, 449, 492] [55, 193, 284, 450, 493] [48, 194, 285, 451, 494] [49, 195, 286, 452, 495] [50, 196, 287, 453, 488] [51, 197, 280, 454, 489] [52, 198, 281, 455, 490] [53, 199, 282, 448, 491] [62, 160, 315, 457, 500] [63, 161, 316, 458, 501] [56, 162, 317, 459, 502] [57, 163, 318, 460, 503] [58, 164, 319, 461, 496] [59, 165, 312, 462, 497] [60, 166, 313, 463, 498] [61, 167, 314, 456, 499] [70, 168, 307, 465, 508] [71, 169, 308, 466, 509] [64, 170, 309, 467, 510] [65, 171, 310, 468, 511] [66, 172, 311, 469, 504] [67, 173, 304, 470, 505] [68, 174, 305, 471, 506] [69, 175, 306, 464, 507] [78, 176, 299, 473, 516] [79, 177, 300, 474, 517] [72, 178, 301, 475, 518] [73, 179, 302, 476, 519] [74, 180, 303, 477, 512] [75, 181, 296, 478, 513] [76, 182, 297, 479, 514] [77, 183, 298, 472, 515] [86, 224, 331, 466, 482] [87, 225, 332, 459, 483] [80, 226, 333, 468, 484] [81, 227, 334, 461, 485] [82, 228, 335, 470, 486] [83, 229, 328, 463, 487] [84, 230, 329, 464, 480] [85, 231, 330, 457, 481] [94, 232, 323, 474, 490] [95, 233, 324, 467, 491] [88, 234, 325, 476, 492] [89, 235, 326, 469, 493] [90, 236, 327, 478, 494] [91, 237, 320, 471, 495] [92, 238, 321, 472, 488] [93, 239, 322, 465, 489] [102, 200, 355, 442, 498] [103, 201, 356, 475, 499] [96, 202, 357, 444, 500] [97, 203, 358, 477, 501] [98, 204, 359, 446, 502] [99, 205, 352, 479, 503] [100, 206, 353, 440, 496] [101, 207, 354, 473, 497] [110, 208, 347, 450, 506] [111, 209, 348, 443, 507] [104, 210, 349, 452, 508] [105, 211, 350, 445, 509] [106, 212, 351, 454, 510] [107, 213, 344, 447, 511] [108, 214, 345, 448, 504] [109, 215, 346, 441, 505] [118, 216, 339, 458, 514] [119, 217, 340, 451, 515] [112, 218, 341, 460, 516] [113, 219, 342, 453, 517] [114, 220, 343, 462, 518] [115, 221, 336, 455, 519] [116, 222, 337, 456, 512] [117, 223, 338, 449, 513]
H_Z (240 checks, sparse supports)
[14, 47, 102, 380, 442] [39, 40, 111, 381, 443] [8, 41, 96, 382, 444] [33, 42, 105, 383, 445] [10, 43, 98, 376, 446] [35, 44, 107, 377, 447] [12, 45, 100, 378, 440] [37, 46, 109, 379, 441] [22, 55, 110, 388, 450] [7, 48, 119, 389, 451] [16, 49, 104, 390, 452] [1, 50, 113, 391, 453] [18, 51, 106, 384, 454] [3, 52, 115, 385, 455] [20, 53, 108, 386, 448] [5, 54, 117, 387, 449] [30, 63, 118, 396, 458] [15, 56, 87, 397, 459] [24, 57, 112, 398, 460] [9, 58, 81, 399, 461] [26, 59, 114, 392, 462] [11, 60, 83, 393, 463] [28, 61, 116, 394, 456] [13, 62, 85, 395, 457] [38, 71, 86, 364, 466] [23, 64, 95, 365, 467] [32, 65, 80, 366, 468] [17, 66, 89, 367, 469] [34, 67, 82, 360, 470] [19, 68, 91, 361, 471] [36, 69, 84, 362, 464] [21, 70, 93, 363, 465] [6, 79, 94, 372, 474] [31, 72, 103, 373, 475] [0, 73, 88, 374, 476] [25, 74, 97, 375, 477] [2, 75, 90, 368, 478] [27, 76, 99, 369, 479] [4, 77, 92, 370, 472] [29, 78, 101, 371, 473] [12, 44, 86, 420, 482] [37, 45, 87, 421, 483] [14, 46, 80, 422, 484] [39, 47, 81, 423, 485] [8, 40, 82, 416, 486] [33, 41, 83, 417, 487] [10, 42, 84, 418, 480] [35, 43, 85, 419, 481] [20, 52, 94, 428, 490] [5, 53, 95, 429, 491] [22, 54, 88, 430, 492] [7, 55, 89, 431, 493] [16, 48, 90, 424, 494] [1, 49, 91, 425, 495] [18, 50, 92, 426, 488] [3, 51, 93, 427, 489] [28, 60, 102, 436, 498] [13, 61, 103, 437, 499] [30, 62, 96, 438, 500] [15, 63, 97, 439, 501] [24, 56, 98, 432, 502] [9, 57, 99, 433, 503] [26, 58, 100, 434, 496] [11, 59, 101, 435, 497] [36, 68, 110, 404, 506] [21, 69, 111, 405, 507] [38, 70, 104, 406, 508] [23, 71, 105, 407, 509] [32, 64, 106, 400, 510] [17, 65, 107, 401, 511] [34, 66, 108, 402, 504] [19, 67, 109, 403, 505] [4, 76, 118, 412, 514] [29, 77, 119, 413, 515] [6, 78, 112, 414, 516] [31, 79, 113, 415, 517] [0, 72, 114, 408, 518] [25, 73, 115, 409, 519] [2, 74, 116, 410, 512] [27, 75, 117, 411, 513] [134, 167, 222, 375, 456] [159, 160, 231, 368, 457] [128, 161, 216, 369, 458] [153, 162, 225, 370, 459] [130, 163, 218, 371, 460] [155, 164, 227, 372, 461] [132, 165, 220, 373, 462] [157, 166, 229, 374, 463] [142, 175, 230, 367, 464] [127, 168, 239, 360, 465] [136, 169, 224, 361, 466] [121, 170, 233, 362, 467] [138, 171, 226, 363, 468] [123, 172, 235, 364, 469] [140, 173, 228, 365, 470] [125, 174, 237, 366, 471] [150, 183, 238, 399, 472] [135, 176, 207, 392, 473] [144, 177, 232, 393, 474] [129, 178, 201, 394, 475] [146, 179, 234, 395, 476] [131, 180, 203, 396, 477] [148, 181, 236, 397, 478] [133, 182, 205, 398, 479] [158, 191, 206, 391, 440] [143, 184, 215, 384, 441] [152, 185, 200, 385, 442] [137, 186, 209, 386, 443] [154, 187, 202, 387, 444] [139, 188, 211, 388, 445] [156, 189, 204, 389, 446] [141, 190, 213, 390, 447] [126, 199, 214, 383, 448] [151, 192, 223, 376, 449] [120, 193, 208, 377, 450] [145, 194, 217, 378, 451] [122, 195, 210, 379, 452] [147, 196, 219, 380, 453] [124, 197, 212, 381, 454] [149, 198, 221, 382, 455] [132, 164, 206, 415, 496] [157, 165, 207, 408, 497] [134, 166, 200, 409, 498] [159, 167, 201, 410, 499] [128, 160, 202, 411, 500] [153, 161, 203, 412, 501] [130, 162, 204, 413, 502] [155, 163, 205, 414, 503] [140, 172, 214, 407, 504] [125, 173, 215, 400, 505] [142, 174, 208, 401, 506] [127, 175, 209, 402, 507] [136, 168, 210, 403, 508] [121, 169, 211, 404, 509] [138, 170, 212, 405, 510] [123, 171, 213, 406, 511] [148, 180, 222, 439, 512] [133, 181, 223, 432, 513] [150, 182, 216, 433, 514] [135, 183, 217, 434, 515] [144, 176, 218, 435, 516] [129, 177, 219, 436, 517] [146, 178, 220, 437, 518] [131, 179, 221, 438, 519] [156, 188, 230, 431, 480] [141, 189, 231, 424, 481] [158, 190, 224, 425, 482] [143, 191, 225, 426, 483] [152, 184, 226, 427, 484] [137, 185, 227, 428, 485] [154, 186, 228, 429, 486] [139, 187, 229, 430, 487] [124, 196, 238, 423, 488] [149, 197, 239, 416, 489] [126, 198, 232, 417, 490] [151, 199, 233, 418, 491] [120, 192, 234, 419, 492] [145, 193, 235, 420, 493] [122, 194, 236, 421, 494] [147, 195, 237, 422, 495] [254, 287, 342, 370, 453] [279, 280, 351, 371, 454] [248, 281, 336, 372, 455] [273, 282, 345, 373, 448] [250, 283, 338, 374, 449] [275, 284, 347, 375, 450] [252, 285, 340, 368, 451] [277, 286, 349, 369, 452] [262, 295, 350, 378, 445] [247, 288, 359, 379, 446] [256, 289, 344, 380, 447] [241, 290, 353, 381, 440] [258, 291, 346, 382, 441] [243, 292, 355, 383, 442] [260, 293, 348, 376, 443] [245, 294, 357, 377, 444] [270, 303, 358, 386, 477] [255, 296, 327, 387, 478] [264, 297, 352, 388, 479] [249, 298, 321, 389, 472] [266, 299, 354, 390, 473] [251, 300, 323, 391, 474] [268, 301, 356, 384, 475] [253, 302, 325, 385, 476] [278, 311, 326, 394, 469] [263, 304, 335, 395, 470] [272, 305, 320, 396, 471] [257, 306, 329, 397, 464] [274, 307, 322, 398, 465] [259, 308, 331, 399, 466] [276, 309, 324, 392, 467] [261, 310, 333, 393, 468] [246, 319, 334, 362, 461] [271, 312, 343, 363, 462] [240, 313, 328, 364, 463] [265, 314, 337, 365, 456] [242, 315, 330, 366, 457] [267, 316, 339, 367, 458] [244, 317, 332, 360, 459] [269, 318, 341, 361, 460] [252, 284, 326, 410, 493] [277, 285, 327, 411, 494] [254, 286, 320, 412, 495] [279, 287, 321, 413, 488] [248, 280, 322, 414, 489] [273, 281, 323, 415, 490] [250, 282, 324, 408, 491] [275, 283, 325, 409, 492] [260, 292, 334, 418, 485] [245, 293, 335, 419, 486] [262, 294, 328, 420, 487] [247, 295, 329, 421, 480] [256, 288, 330, 422, 481] [241, 289, 331, 423, 482] [258, 290, 332, 416, 483] [243, 291, 333, 417, 484] [268, 300, 342, 426, 517] [253, 301, 343, 427, 518] [270, 302, 336, 428, 519] [255, 303, 337, 429, 512] [264, 296, 338, 430, 513] [249, 297, 339, 431, 514] [266, 298, 340, 424, 515] [251, 299, 341, 425, 516] [276, 308, 350, 434, 509] [261, 309, 351, 435, 510] [278, 310, 344, 436, 511] [263, 311, 345, 437, 504] [272, 304, 346, 438, 505] [257, 305, 347, 439, 506] [274, 306, 348, 432, 507] [259, 307, 349, 433, 508] [244, 316, 358, 402, 501] [269, 317, 359, 403, 502] [246, 318, 352, 404, 503] [271, 319, 353, 405, 496] [240, 312, 354, 406, 497] [265, 313, 355, 407, 498] [242, 314, 356, 400, 499] [267, 315, 357, 401, 500]
Code ID 520-44-8 · download JSON · raw on GitHub