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[[330,66,10]] d ≤
n
330
k
66
d
10
kd²/n
20.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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)
[166, 168, 170, 178, 201, 209, 227, 229, 232, 236]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[81, 103, 107, 129, 208, 220, 235, 240, 248, 261]
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)×132 (2,2)×132 (3,2)×132 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 132 (1,3): 132 (1,5): 66 (2,2): 132 (2,3): 825 (2,4): 396 (2,5): 594 (2,6): 990 (2,8): 264 (3,2): 132 (3,3): 2574 (3,4): 4741 (3,5): 4884 (3,6): 13827 (3,7): 10197 (3,8): 6138 (3,9): 9306 (3,10): 396 (3,11): 2442 (3,13): 66
trapping sets H_Z (1,2)×132 (2,2)×132 (3,2)×132 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 132 (1,3): 132 (1,5): 66 (2,2): 132 (2,3): 792 (2,4): 429 (2,5): 660 (2,6): 924 (2,8): 264 (3,2): 132 (3,3): 2376 (3,4): 4664 (3,5): 5830 (3,6): 13959 (3,7): 9361 (3,8): 6633 (3,9): 8580 (3,10): 528 (3,11): 2376 (3,13): 66

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 = C11xD3 (order 66; Cayley table from research/kit/nonabelian_lp.small_nonabelian_groups). Base matrices A = [{0,13,51}, {0,44}] (entries act by the left regular representation L(g)[gh,h]=1) and B = [{0,41,61}, {0,8}] (entries act by the right regular representation R(g)[h,hg]=1). Element index lists: A = [[[0, 13, 51], [0, 44]]], B = [[[0, 41, 61], [0, 8]]]. 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 [10, 10, 10], 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

[[330,66,10]] non-abelian lifted product over C11xD3, 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 = 330 the target was d = 10 with k = 66, which dominates the board's 330-66-9 (same n and k, one more unit of distance, same check weight).

What was searched

Batch 1, order 66: 3000 random (3,2)/(3,2) codes over the six order-66 presentations (ZSZ(11,6,10), ZSZ(33,2,10), ZSZ(33,2,23), ZSZ(33,2,32), C3xD11, C11xD3), weight-2 entries restricted to elements of order at least 10, seed 11: d histogram 4:92, 5:2515, 6:42, 7:127, 8:187, 9:12, no d = 10. Only C11xD3 and ZSZ(33,2,23) (both Z11 x S3) produced d >= 6; D33, Z3 x D11 and ZSZ(11,6,10) stayed at d <= 5. Batch 2, order 66, focused on C11xD3: 3000 codes, same filter, seed 21: d histogram 4:59, 5:1781, 6:108, 7:364, 8:628, 9:45, 10:6. Two of the six d = 10 codes were laddered and both 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 -> 10, 5k -> 10, 50k -> 10, 200k per side (packaging) -> X 10, Z 10. Gate verdict: passed = True; labels: advances the weight-8 x unrestricted board; literature novelty UNVERIFIED. Claim: witness-backed upper bound d <= 10 (confidence upper_bound), k = 66 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 per-order screen histograms of this run are in the search section above; no ladder for the packaged code collapsed. 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(66, 66))["C11xD3"]
A = [[[0, 13, 51], [0, 44]]]
B = [[[0, 41, 61], [0, 8]]]
HX, HZ = lifted_product_base(mul, A, B)   # [[330,66,10]], check weight 8

Parity checks

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