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[[360,74,8]] d ≤
n
360
k
74
d
8
kd²/n
13.156
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[181, 193, 224, 228, 237, 265, 268, 275]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[42, 54, 147, 152, 177, 182, 197, 198]
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 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): 432 (2,5): 720 (2,6): 1080 (2,8): 288 (3,2): 144 (3,3): 2784 (3,4): 4800 (3,5): 5400 (3,6): 15936 (3,7): 10800 (3,8): 7416 (3,9): 10440 (3,10): 576 (3,11): 2448 (3,13): 72
trapping sets H_Z (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_Z
(1,2): 144 (1,3): 144 (1,5): 72 (2,2): 144 (2,3): 864 (2,4): 432 (2,5): 720 (2,6): 1080 (2,8): 288 (3,2): 144 (3,3): 2736 (3,4): 4848 (3,5): 5472 (3,6): 15744 (3,7): 11088 (3,8): 7560 (3,9): 10248 (3,10): 576 (3,11): 2376 (3,13): 72

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
notes Checked against the board: not equivalent to any existing entry (gate dedup clean).
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

[[360,74,8]] — non-abelian lifted product over C12×D3, check weight 8

Direction & hypothesis

Hackathon (issue #1155) weight-8 cell. The 2026-09-16 girth-cap fieldnote (fieldnotes/2026-09-16-lifted-product-girth-cap.md) capped one-row non-abelian lifted products with a weight-2 entry at d <= 8 for |G| <= 140, but left the (3,2)/(3,2) profile (check weight 8, rate 1/5) reaching d = 9 at n = 350, 390. A frontier scan showed d >= 8 at n = 5|G| for |G| in {63,64,66,68,72,74,76} are all board-advancing. This is a d = 8 point at |G| = 72 (n = 360) with k = 74 (higher than the nominal |G| = 72), on the non-metacyclic direct product C12×D3.

What was searched

The (3,2)/(3,2) profile over all ZSZ presentations and small non-abelian groups of orders 63, 64, 66, 68, 72, 74, 76 (49 groups). ~18,000 random codes screened at 400 RIS trials (gf2_fast), survivors re-screened at 5k-10k. At n = 360 only d = 8 codes were found (no d >= 9); this is one of several [[360,74,8]] codes.

Evidence trail

[[360,74,8]] on C12×D3, A = [[[0,26,56],[0,71]]], B = [[[0,8,46],[0,35]]]. CSS holds, k = 74, max check weight 8. Fast RIS: d <= 8 at 2k trials, flat at 10k. validate_candidate: passed, board_advancing true, no lighter logical in the gate's RIS search, no WL-equivalent board entry. Witness-backed upper bound (confidence: upper_bound). Literature novelty UNVERIFIED.

Dead ends

  • The girth-cap fieldnote's two d=9 survivors ([[350,70,9]], [[390,78,9]]) are

    • already on the board (duplicates).

  • No d >= 9 at n = 360; d = 8 is the achievable ceiling there.
  • No d >= 8 at |G| = 68, 74, 76 (n = 340, 370, 380).

Tools

DeepSeek V4 Flash 0731 (this agent), research/kit/nonabelian_lp.py, research/kit/surrogate.py distance_rand (gf2_fast backend), verify/validate_candidate.py.

Reproduction

import sys; sys.path.insert(0, "research/kit"); sys.path.insert(0, "verify")
from nonabelian_lp import group_from_name, lifted_product_base
mul = group_from_name("C12xD3")
A = [[[0, 26, 56], [0, 71]]]
B = [[[0, 8, 46], [0, 35]]]
HX, HZ = lifted_product_base(mul, A, B)   # [[360,74,8]], check weight 8

Parity checks

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