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[[360,72,9]] d ≤
n
360
k
72
d
9
kd²/n
16.2
w
8
X/Z
1.11

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Distance

X/Z asymmetry 1.11 · d_X ≤ 10, d_Z ≤ 9 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[7, 15, 17, 19, 87, 91, 107, 111, 126, 134]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[73, 79, 104, 107, 113, 136, 239, 251, 263]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7–8 (mean 7.5) · H_Z 7–8 (mean 7.5)
qubit degrees H_X 2–5 (mean 3.0) · H_Z 2–5 (mean 3.0)
trapping sets H_X (1,2)×144 (2,2)×144 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 144 (1,3): 144 (1,5): 72 (2,2): 144 (2,3): 864 (2,4): 480 (2,5): 720 (2,6): 984 (2,8): 288 (3,2): 144 (3,3): 2736 (3,4): 5136 (3,5): 5904 (3,6): 15024 (3,7): 10440 (3,8): 7512 (3,9): 9048 (3,10): 576 (3,11): 2256 (3,13): 72
trapping sets H_Z (1,2)×144 (2,2)×144 (3,2)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 144 (1,3): 144 (1,5): 72 (2,2): 144 (2,3): 864 (2,4): 480 (2,5): 720 (2,6): 984 (2,8): 288 (3,2): 288 (3,3): 2592 (3,4): 4800 (3,5): 6336 (3,6): 14736 (3,7): 10344 (3,8): 7512 (3,9): 9336 (3,10): 576 (3,11): 2256 (3,13): 72

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Lifted product over the non-abelian group algebra F_2[G], G = C12xD3 (order 72; Cayley table from research/kit/nonabelian_lp.small_nonabelian_groups). Base matrices A = [{0,6,33}, {0,44}] (entries act by the left regular representation L(g)[gh,h]=1) and B = [{0,22,47}, {0,68}] (entries act by the right regular representation R(g)[h,hg]=1). Element index lists: A = [[[0, 6, 33], [0, 44]]], B = [[[0, 22, 47], [0, 68]]]. Qubit blocks of size |G|: sector 1 holds (i,j) for i in cols(A), j in cols(B) at block i*n_B+j; sector 2 holds (r,s) for r in rows(A), s in rows(B). X-check (r,j) = [L(A[r][i]) on (i,j)] + [R(B[s][j]) on (r,s)]; Z-check (i,s) = [R(B[s][j])^T on (i,j)] + [L(A[r][i])^T on (r,s)]. Same construction as the weight-9 mitten / ZSZ-LP codes of arXiv:2607.28795 and arXiv:2607.27644 (four weight-3 entries there), here with entry weights (3,2)/(3,2), check weight 8. Built with research/kit/nonabelian_lp.lifted_product_base.
model Claude Claude Fable 5.1 (claimed, not verified)
date 2026-09-20
notes Distance is a witness-backed upper bound from randomized information-set search (research/kit/surrogate.distance_rand, gf2_fast backend); screen 400 trials, ladder [9, 9, 9], packaging 200000 trials per side. Advances the unrestricted weight-8 board cell; novelty vs the literature unverified.
family lifted product (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[360,72,9]] non-abelian lifted product over C12xD3, check weight 8

Direction & hypothesis

Unrestricted weight-8 cell, rate-1/5 band at n = 315 to 390. The one-row lifted product with entry weights (3,2)/(3,2) is the only profile below check weight 9 that escapes the Cayley-graph girth cap (fieldnotes/2026-09-16-lifted-product-girth-cap.md). The Sep-16 and Sep-18 sweeps placed d = 8 to 10 points at n = 315, 320, 330, 350, 360 and 390 with about 20 to 120 random codes per group. Hypothesis: a denser sweep of the orders 63 to 78, with the weight-2 entries 1 + g restricted to elements of order at least the target distance (the sum 1 + g + ... + g^(ord(g)-1) is a seed codeword of weight ord(g)), reaches one distance unit higher at the same n. At n = 360 the frontier holds 360-74-8 and the board also has a non-frontier 360-72-8; a d = 9 code with k = 72 joins the frontier (nothing on the board has n <= 360, k >= 72, d >= 9 at check weight <= 8) and dominates 360-72-8, while leaving 360-74-8 and 350-70-9 in place. It fills a gap and displaces no frontier entry.

What was searched

Batch 2, order 72: 1500 random (3,2)/(3,2) codes over the 16 non-dihedral order-72 presentations (nine ZSZ(l1,l2,q) and C3xS4, C6xA4, C2xD18, C3xD12, C4xD9, C6xD6, C9xD4, C12xD3), weight-2 entries restricted to elements of order at least 9, seed 25. Screen histogram of d at 400 trials: 4:88, 5:1226, 6:46, 7:43, 8:87, 9:1. The single d = 9 code sits on C12xD3 (Z12 x S3), the group of the board's 360-74-8; it was laddered and held. Screen: research/kit/search.screen with sample_nonabelian_lp, 400 fast RIS trials (gf2_fast), two threads. Board pre-check against codes/*.json under the site's Pareto rule (n, k, d, w). Ladder on the survivors: 5k then 50k fast RIS trials. Packaging: 200k fast RIS trials per side, witness installed if lighter. Gate: verify/validate_candidate.py.

Evidence trail

Ladder for this code (trials -> lightest logical found): 400 -> 9, 5k -> 9, 50k -> 9, 200k per side (packaging) -> X 10, Z 9. Gate verdict: passed = True; labels: advances the weight-8 x unrestricted board; literature novelty UNVERIFIED. Claim: witness-backed upper bound d <= 9 (confidence upper_bound), k = 72 exactly (n - rank H_X - rank H_Z), max check weight 8. Advances the unrestricted weight-8 board; novelty vs the literature unverified.

Dead ends

The sweep counts and ladders are in the search section above. Structural dead ends of the family are in fieldnotes/2026-09-16-lifted-product-girth-cap.md (all-weight-2 sides, rate 2/5, 2x3 monomial bases).

Tools

Claude Fable 5.1 (Claude Code, unattended workflow direction lp-w8), research/kit/nonabelian_lp.py, research/kit/search.py, research/kit/surrogate.py with the gf2_fast backend, verify/validate_candidate.py. Two CPU threads, under 80 minutes wall clock for the whole direction.

Reproduction

import sys; sys.path.insert(0, "research/kit"); sys.path.insert(0, "verify")
from nonabelian_lp import zsz, small_nonabelian_groups, lifted_product_base
mul = dict(small_nonabelian_groups(72, 72))["C12xD3"]
A = [[[0, 6, 33], [0, 44]]]
B = [[[0, 22, 47], [0, 68]]]
HX, HZ = lifted_product_base(mul, A, B)   # [[360,72,9]], check weight 8

Parity checks

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