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[[336,24,24]] d ≤
n
336
k
24
d
24
kd²/n
41.143
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 24, d_Z ≤ 24 · 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 24 · witness weight 24 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified) · found at 3×108 trials · survived 3×108 trials · 2026-09-25
witness operator (support, 24 qubits)
[6, 9, 20, 23, 34, 37, 44, 46, 58, 60, 72, 74, 84, 85, 98, 99, 112, 113, 126, 140, 154, 215, 229, 243]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[11, 19, 32, 40, 52, 55, 73, 76, 100, 121, 143, 164, 187, 208, 221, 242, 254, 259, 264, 275, 280, 285, 311, 332]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×336 (2,6)×4704 (3,6)×2184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 336 (2,6): 4704 (3,6): 2184 (3,8): 92232 (3,10): 9408
trapping sets H_Z (1,4)×336 (2,6)×4704 (3,6)×2184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 336 (2,6): 4704 (3,6): 2184 (3,8): 92232 (3,10): 9408

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Free-action lifted product over Z_4 x Z_42 with the published supports from arXiv:2608.08996v1.
model human (claimed, not verified)
date 2026-08-28
notes Reconstructed from the paper Supplementary Information; 20,000 pure-Python RIS trials per side at seed 20260828 found no lighter logical.
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

[[336,24,24]]: distance revision of the board's [[336,24,24]] entry

Revision history

The entry keeps its parameters [[336,24,24]]. The code, its checks, its layout, and its original provenance are unchanged; only the distance block is corrected. A GPU random-information-set search (verify/ris_gpu.py, recover mode, 300,000,000 trials per side, pair depth 0, seed 4101) exhibits a weight-24 X logical, so the previous witness-backed bound was overstated on that side; the overall distance d = 24 is unchanged. Each lighter witness is carried in the entry with the budget it was found at; the other side keeps its original witness where it was not refuted. Distance remains an upper bound, not an exact claim.

Evidence

Every GPU proposal was re-validated on the CPU against the committed check matrices (in the kernel of the opposite side's checks, outside the row space of its own side) before it was recorded. The pass is the weekly fresh-seed re-measurement of the board's cell leaders (issue 2025, run of 2026-09-24); every recorded reading is the lightest CPU-verified logical at that budget.

| side | seed | claimed | GPU best weight | CPU-verified witness | trials | |---|---|---|---|---|---| | X | 4101 | 26 | 24 | 24 | 300,000,000 | | Z | 4101 | 24 | 24 | 24 | 300,000,000 | | X | 4102 | 26 | 24 | 24 | 300,000,000 | | Z | 4102 | 24 | 24 | 24 | 300,000,000 |

Original note, with the file paths updated

[[336,24,24]] — abelian lifted-product reconstruction

Direction & hypothesis

Targeted the unrestricted weight-8 frontier using a directly reproducible row from arXiv:2608.08996v1.

What was searched

No parameter sweep: rebuild the paper's free-action row over Z_4 x Z_42.

Evidence trail

The matrices recompute to n=336, k=24, max check weight 8, and CSS commutation. The paper reports d<=24. Pure-Python RIS found no lighter logical than 24 in 20,000 trials per side (seed 20260828). The claim remains a witness-backed upper bound, not an exact proof.

Dead ends

None for this reconstruction.

Tools

Python 3, the repository group-algebra constructor, submit builder, and trusted verifier.

Reproduction

Over Z_4 x Z_42, use A={(0,0),(1,2),(2,5),(3,6)} and B={(0,0),(1,13),(2,9),(3,19)}, from the Supplementary Information of arXiv:2608.08996v1.

Parity checks

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