[[390,78,10]] · QEC Challenge

[[390,78,10]] d ≤

390

kd²/n

20.0

X/Z

Share this result

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)

[4, 40, 48, 74, 82, 114, 121, 122, 152, 155]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[91, 105, 131, 145, 234, 243, 247, 274, 281, 287]

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)×156 (2,2)×156 (3,2)×156 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 156 (1,3): 156 (1,5): 78 (2,2): 156 (2,3): 975 (2,4): 559 (2,5): 702 (2,6): 988 (2,8): 312 (3,2): 156 (3,3): 3068 (3,4): 6279 (3,5): 6695 (3,6): 14625 (3,7): 10387 (3,8): 6916 (3,9): 8736 (3,10): 468 (3,11): 2600 (3,13): 78

trapping sets H_Z (1,2)×156 (2,2)×156 (3,2)×156 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

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 = Z_39 x|_14 Z_2 (order 78; generators x, y with x³⁹ = y² = 1, y x = x¹⁴ y; element x^a y^b has index a*2+b). Base matrices A = [1 + x¹⁷ + x³³y¹, 1 + x⁷] (entries act by the left regular representation L(g)[gh,h]=1) and B = [1 + x²⁰ + x³⁶y¹, 1 + x²²] (entries act by the right regular representation R(g)[h,hg]=1). Element index lists: A = [[[0, 34, 67], [0, 14]]], B = [[[0, 40, 73], [0, 44]]]. 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 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 300000 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

[[390,78,10]] non-abelian lifted product over ZSZ(39,2,14), 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 = 390 the target was d = 10 with k = 78, which dominates the board's 390-78-9 (same n and k, one more unit of distance, same check weight).

What was searched

Batch 1, order 78: 3000 random (3,2)/(3,2) codes over the eleven order-78 presentations (five ZSZ(13,6,q), two ZSZ(26,3,q), ZSZ(39,2,14), ZSZ(39,2,25), ZSZ(39,2,38), C3xD13), weight-2 entries restricted to elements of order at least 10, seed 13. Screen histogram of d at 400 trials: 4:282, 5:2522, 6:4, 7:23, 8:66, 9:4, 10:3. Every d >= 6 code sits on ZSZ(39,2,14), which is Z13 x S3; the other ten presentations (Z3 x D13 twice, D39, the ZSZ(13,6,q) and ZSZ(26,3,q) groups) never exceeded d = 5 in about 270 codes each. 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: 300k 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, 300k 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 = 78 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 = zsz(39, 2, 14)
A = [[[0, 34, 67], [0, 14]]]
B = [[[0, 40, 73], [0, 44]]]
HX, HZ = lifted_product_base(mul, A, B)   # [[390,78,10]], check weight 8

Parity checks

X-checks 156 (max weight 8) · Z-checks 156 (max weight 8)

H_X (156 checks, sparse supports)

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

H_Z (156 checks, sparse supports)

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

Code ID 390-78-10 · download JSON · raw on GitHub