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[[315,63,8]] d ≤
n
315
k
63
d
8
kd²/n
12.8
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)
[129, 138, 191, 192, 216, 231, 248, 251]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[6, 57, 131, 132, 141, 158, 170, 182]
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)×126 (2,2)×126 (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): 126 (1,3): 126 (1,5): 63 (2,2): 126 (2,3): 777 (2,4): 396 (2,5): 588 (2,6): 909 (2,8): 252 (3,2): 132 (3,3): 2499 (3,4): 4545 (3,5): 4698 (3,6): 13122 (3,7): 9369 (3,8): 5976 (3,9): 8691 (3,10): 462 (3,11): 1944 (3,13): 63
trapping sets H_Z (1,2)×126 (2,2)×126 (3,2)×258 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 126 (1,3): 126 (1,5): 63 (2,2): 126 (2,3): 777 (2,4): 396 (2,5): 588 (2,6): 909 (2,8): 252 (3,2): 258 (3,3): 2478 (3,4): 4167 (3,5): 4728 (3,6): 13143 (3,7): 9501 (3,8): 5913 (3,9): 8529 (3,10): 462 (3,11): 2133 (3,13): 63

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

[[315,63,8]] — non-abelian lifted product over ZSZ(21,3,16), 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 the d = 8 point at |G| = 63 (n = 315).

What was searched

The (3,2)/(3,2) profile over the ZSZ presentations of order 63 (ZSZ(21,3,4), ZSZ(21,3,16)) plus orders 68, 74, 76. 6000 random codes screened at 400 RIS trials (gf2_fast), survivors re-screened at 10k. Only one d >= 8 code was found across these four orders: this [[315,63,8]] on ZSZ(21,3,16). Orders 68, 74, 76 yielded no d >= 8 code.

Evidence trail

[[315,63,8]] on ZSZ(21,3,16), A = [[[0,9,44],[0,51]]], B = [[[0,39,56],[0,48]]]. CSS holds, k = 63, 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

  • No d >= 8 at |G| = 68, 74, 76 (n = 340, 370, 380) in 6000 candidates each.
  • At |G| = 63, only ZSZ(21,3,16) produced a d = 8 code; ZSZ(21,3,4) did not.

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 zsz, lifted_product_base
mul = zsz(21, 3, 16)
A = [[[0, 9, 44], [0, 51]]]
B = [[[0, 39, 56], [0, 48]]]
HX, HZ = lifted_product_base(mul, A, B)   # [[315,63,8]], check weight 8

Parity checks

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