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[[336,28,20]] d ≤
n
336
k
28
d
20
kd²/n
33.333
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · 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 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[15, 43, 44, 47, 93, 108, 124, 137, 160, 177, 179, 204, 212, 223, 250, 262, 290, 297, 309, 327]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[28, 66, 111, 148, 175, 179, 201, 209, 211, 215, 237, 245, 253, 275, 283, 291, 308, 316, 324, 328]
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)×2352 (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): 2352 (3,8): 91728 (3,10): 9408
trapping sets H_Z (1,4)×336 (2,6)×4704 (3,6)×2352 (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): 2352 (3,8): 91728 (3,10): 9408

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Free-action lifted product over Z_2 x Z_2 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

[[336,28,20]] — 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_2 x Z_2 x Z_42.

Evidence trail

The matrices recompute to n=336, k=28, max check weight 8, and CSS commutation. The paper reports d<=20. Pure-Python RIS found no lighter logical than 20 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_2 x Z_2 x Z_42, use A={(0,0,0),(0,1,5),(1,0,2),(1,1,6)} and B={(0,0,0),(0,1,9),(1,0,13),(1,1,19)}, 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, 79, 124, 162, 168, 243, 281, 317] [1, 80, 125, 163, 169, 244, 282, 318] [2, 81, 84, 164, 170, 245, 283, 319] [3, 82, 85, 165, 171, 246, 284, 320] [4, 83, 86, 166, 172, 247, 285, 321] [5, 42, 87, 167, 173, 248, 286, 322] [6, 43, 88, 126, 174, 249, 287, 323] [7, 44, 89, 127, 175, 250, 288, 324] [8, 45, 90, 128, 176, 251, 289, 325] [9, 46, 91, 129, 177, 210, 290, 326] [10, 47, 92, 130, 178, 211, 291, 327] [11, 48, 93, 131, 179, 212, 292, 328] [12, 49, 94, 132, 180, 213, 293, 329] [13, 50, 95, 133, 181, 214, 252, 330] [14, 51, 96, 134, 182, 215, 253, 331] [15, 52, 97, 135, 183, 216, 254, 332] [16, 53, 98, 136, 184, 217, 255, 333] [17, 54, 99, 137, 185, 218, 256, 334] [18, 55, 100, 138, 186, 219, 257, 335] [19, 56, 101, 139, 187, 220, 258, 294] [20, 57, 102, 140, 188, 221, 259, 295] [21, 58, 103, 141, 189, 222, 260, 296] [22, 59, 104, 142, 190, 223, 261, 297] [23, 60, 105, 143, 191, 224, 262, 298] [24, 61, 106, 144, 192, 225, 263, 299] [25, 62, 107, 145, 193, 226, 264, 300] [26, 63, 108, 146, 194, 227, 265, 301] [27, 64, 109, 147, 195, 228, 266, 302] [28, 65, 110, 148, 196, 229, 267, 303] [29, 66, 111, 149, 197, 230, 268, 304] [30, 67, 112, 150, 198, 231, 269, 305] [31, 68, 113, 151, 199, 232, 270, 306] [32, 69, 114, 152, 200, 233, 271, 307] [33, 70, 115, 153, 201, 234, 272, 308] [34, 71, 116, 154, 202, 235, 273, 309] [35, 72, 117, 155, 203, 236, 274, 310] [36, 73, 118, 156, 204, 237, 275, 311] [37, 74, 119, 157, 205, 238, 276, 312] [38, 75, 120, 158, 206, 239, 277, 313] [39, 76, 121, 159, 207, 240, 278, 314] [40, 77, 122, 160, 208, 241, 279, 315] [41, 78, 123, 161, 209, 242, 280, 316] [37, 42, 120, 166, 201, 210, 275, 323] [38, 43, 121, 167, 202, 211, 276, 324] [39, 44, 122, 126, 203, 212, 277, 325] [40, 45, 123, 127, 204, 213, 278, 326] [41, 46, 124, 128, 205, 214, 279, 327] [0, 47, 125, 129, 206, 215, 280, 328] [1, 48, 84, 130, 207, 216, 281, 329] [2, 49, 85, 131, 208, 217, 282, 330] [3, 50, 86, 132, 209, 218, 283, 331] [4, 51, 87, 133, 168, 219, 284, 332] [5, 52, 88, 134, 169, 220, 285, 333] [6, 53, 89, 135, 170, 221, 286, 334] [7, 54, 90, 136, 171, 222, 287, 335] [8, 55, 91, 137, 172, 223, 288, 294] [9, 56, 92, 138, 173, 224, 289, 295] [10, 57, 93, 139, 174, 225, 290, 296] [11, 58, 94, 140, 175, 226, 291, 297] [12, 59, 95, 141, 176, 227, 292, 298] [13, 60, 96, 142, 177, 228, 293, 299] [14, 61, 97, 143, 178, 229, 252, 300] [15, 62, 98, 144, 179, 230, 253, 301] [16, 63, 99, 145, 180, 231, 254, 302] [17, 64, 100, 146, 181, 232, 255, 303] [18, 65, 101, 147, 182, 233, 256, 304] [19, 66, 102, 148, 183, 234, 257, 305] [20, 67, 103, 149, 184, 235, 258, 306] [21, 68, 104, 150, 185, 236, 259, 307] [22, 69, 105, 151, 186, 237, 260, 308] [23, 70, 106, 152, 187, 238, 261, 309] [24, 71, 107, 153, 188, 239, 262, 310] [25, 72, 108, 154, 189, 240, 263, 311] [26, 73, 109, 155, 190, 241, 264, 312] [27, 74, 110, 156, 191, 242, 265, 313] [28, 75, 111, 157, 192, 243, 266, 314] [29, 76, 112, 158, 193, 244, 267, 315] [30, 77, 113, 159, 194, 245, 268, 316] [31, 78, 114, 160, 195, 246, 269, 317] [32, 79, 115, 161, 196, 247, 270, 318] [33, 80, 116, 162, 197, 248, 271, 319] [34, 81, 117, 163, 198, 249, 272, 320] [35, 82, 118, 164, 199, 250, 273, 321] [36, 83, 119, 165, 200, 251, 274, 322] [40, 78, 84, 163, 197, 233, 252, 327] [41, 79, 85, 164, 198, 234, 253, 328] [0, 80, 86, 165, 199, 235, 254, 329] [1, 81, 87, 166, 200, 236, 255, 330] [2, 82, 88, 167, 201, 237, 256, 331] [3, 83, 89, 126, 202, 238, 257, 332] [4, 42, 90, 127, 203, 239, 258, 333] [5, 43, 91, 128, 204, 240, 259, 334] [6, 44, 92, 129, 205, 241, 260, 335] [7, 45, 93, 130, 206, 242, 261, 294] [8, 46, 94, 131, 207, 243, 262, 295] [9, 47, 95, 132, 208, 244, 263, 296] [10, 48, 96, 133, 209, 245, 264, 297] [11, 49, 97, 134, 168, 246, 265, 298] [12, 50, 98, 135, 169, 247, 266, 299] [13, 51, 99, 136, 170, 248, 267, 300] [14, 52, 100, 137, 171, 249, 268, 301] [15, 53, 101, 138, 172, 250, 269, 302] [16, 54, 102, 139, 173, 251, 270, 303] [17, 55, 103, 140, 174, 210, 271, 304] [18, 56, 104, 141, 175, 211, 272, 305] [19, 57, 105, 142, 176, 212, 273, 306] [20, 58, 106, 143, 177, 213, 274, 307] [21, 59, 107, 144, 178, 214, 275, 308] [22, 60, 108, 145, 179, 215, 276, 309] [23, 61, 109, 146, 180, 216, 277, 310] [24, 62, 110, 147, 181, 217, 278, 311] [25, 63, 111, 148, 182, 218, 279, 312] [26, 64, 112, 149, 183, 219, 280, 313] [27, 65, 113, 150, 184, 220, 281, 314] [28, 66, 114, 151, 185, 221, 282, 315] [29, 67, 115, 152, 186, 222, 283, 316] [30, 68, 116, 153, 187, 223, 284, 317] [31, 69, 117, 154, 188, 224, 285, 318] [32, 70, 118, 155, 189, 225, 286, 319] [33, 71, 119, 156, 190, 226, 287, 320] [34, 72, 120, 157, 191, 227, 288, 321] [35, 73, 121, 158, 192, 228, 289, 322] [36, 74, 122, 159, 193, 229, 290, 323] [37, 75, 123, 160, 194, 230, 291, 324] [38, 76, 124, 161, 195, 231, 292, 325] [39, 77, 125, 162, 196, 232, 293, 326] [36, 82, 121, 126, 191, 239, 285, 294] [37, 83, 122, 127, 192, 240, 286, 295] [38, 42, 123, 128, 193, 241, 287, 296] [39, 43, 124, 129, 194, 242, 288, 297] [40, 44, 125, 130, 195, 243, 289, 298] [41, 45, 84, 131, 196, 244, 290, 299] [0, 46, 85, 132, 197, 245, 291, 300] [1, 47, 86, 133, 198, 246, 292, 301] [2, 48, 87, 134, 199, 247, 293, 302] [3, 49, 88, 135, 200, 248, 252, 303] [4, 50, 89, 136, 201, 249, 253, 304] [5, 51, 90, 137, 202, 250, 254, 305] [6, 52, 91, 138, 203, 251, 255, 306] [7, 53, 92, 139, 204, 210, 256, 307] [8, 54, 93, 140, 205, 211, 257, 308] [9, 55, 94, 141, 206, 212, 258, 309] [10, 56, 95, 142, 207, 213, 259, 310] [11, 57, 96, 143, 208, 214, 260, 311] [12, 58, 97, 144, 209, 215, 261, 312] [13, 59, 98, 145, 168, 216, 262, 313] [14, 60, 99, 146, 169, 217, 263, 314] [15, 61, 100, 147, 170, 218, 264, 315] [16, 62, 101, 148, 171, 219, 265, 316] [17, 63, 102, 149, 172, 220, 266, 317] [18, 64, 103, 150, 173, 221, 267, 318] [19, 65, 104, 151, 174, 222, 268, 319] [20, 66, 105, 152, 175, 223, 269, 320] [21, 67, 106, 153, 176, 224, 270, 321] [22, 68, 107, 154, 177, 225, 271, 322] [23, 69, 108, 155, 178, 226, 272, 323] [24, 70, 109, 156, 179, 227, 273, 324] [25, 71, 110, 157, 180, 228, 274, 325] [26, 72, 111, 158, 181, 229, 275, 326] [27, 73, 112, 159, 182, 230, 276, 327] [28, 74, 113, 160, 183, 231, 277, 328] [29, 75, 114, 161, 184, 232, 278, 329] [30, 76, 115, 162, 185, 233, 279, 330] [31, 77, 116, 163, 186, 234, 280, 331] [32, 78, 117, 164, 187, 235, 281, 332] [33, 79, 118, 165, 188, 236, 282, 333] [34, 80, 119, 166, 189, 237, 283, 334] [35, 81, 120, 167, 190, 238, 284, 335]
H_Z (168 checks, sparse supports)
[0, 51, 97, 145, 168, 215, 254, 300] [1, 52, 98, 146, 169, 216, 255, 301] [2, 53, 99, 147, 170, 217, 256, 302] [3, 54, 100, 148, 171, 218, 257, 303] [4, 55, 101, 149, 172, 219, 258, 304] [5, 56, 102, 150, 173, 220, 259, 305] [6, 57, 103, 151, 174, 221, 260, 306] [7, 58, 104, 152, 175, 222, 261, 307] [8, 59, 105, 153, 176, 223, 262, 308] [9, 60, 106, 154, 177, 224, 263, 309] [10, 61, 107, 155, 178, 225, 264, 310] [11, 62, 108, 156, 179, 226, 265, 311] [12, 63, 109, 157, 180, 227, 266, 312] [13, 64, 110, 158, 181, 228, 267, 313] [14, 65, 111, 159, 182, 229, 268, 314] [15, 66, 112, 160, 183, 230, 269, 315] [16, 67, 113, 161, 184, 231, 270, 316] [17, 68, 114, 162, 185, 232, 271, 317] [18, 69, 115, 163, 186, 233, 272, 318] [19, 70, 116, 164, 187, 234, 273, 319] [20, 71, 117, 165, 188, 235, 274, 320] [21, 72, 118, 166, 189, 236, 275, 321] [22, 73, 119, 167, 190, 237, 276, 322] [23, 74, 120, 126, 191, 238, 277, 323] [24, 75, 121, 127, 192, 239, 278, 324] [25, 76, 122, 128, 193, 240, 279, 325] [26, 77, 123, 129, 194, 241, 280, 326] [27, 78, 124, 130, 195, 242, 281, 327] [28, 79, 125, 131, 196, 243, 282, 328] [29, 80, 84, 132, 197, 244, 283, 329] [30, 81, 85, 133, 198, 245, 284, 330] [31, 82, 86, 134, 199, 246, 285, 331] [32, 83, 87, 135, 200, 247, 286, 332] [33, 42, 88, 136, 201, 248, 287, 333] [34, 43, 89, 137, 202, 249, 288, 334] [35, 44, 90, 138, 203, 250, 289, 335] [36, 45, 91, 139, 204, 251, 290, 294] [37, 46, 92, 140, 205, 210, 291, 295] [38, 47, 93, 141, 206, 211, 292, 296] [39, 48, 94, 142, 207, 212, 293, 297] [40, 49, 95, 143, 208, 213, 252, 298] [41, 50, 96, 144, 209, 214, 253, 299] [9, 42, 103, 139, 173, 210, 258, 296] [10, 43, 104, 140, 174, 211, 259, 297] [11, 44, 105, 141, 175, 212, 260, 298] [12, 45, 106, 142, 176, 213, 261, 299] [13, 46, 107, 143, 177, 214, 262, 300] [14, 47, 108, 144, 178, 215, 263, 301] [15, 48, 109, 145, 179, 216, 264, 302] [16, 49, 110, 146, 180, 217, 265, 303] [17, 50, 111, 147, 181, 218, 266, 304] [18, 51, 112, 148, 182, 219, 267, 305] [19, 52, 113, 149, 183, 220, 268, 306] [20, 53, 114, 150, 184, 221, 269, 307] [21, 54, 115, 151, 185, 222, 270, 308] [22, 55, 116, 152, 186, 223, 271, 309] [23, 56, 117, 153, 187, 224, 272, 310] [24, 57, 118, 154, 188, 225, 273, 311] [25, 58, 119, 155, 189, 226, 274, 312] [26, 59, 120, 156, 190, 227, 275, 313] [27, 60, 121, 157, 191, 228, 276, 314] [28, 61, 122, 158, 192, 229, 277, 315] [29, 62, 123, 159, 193, 230, 278, 316] [30, 63, 124, 160, 194, 231, 279, 317] [31, 64, 125, 161, 195, 232, 280, 318] [32, 65, 84, 162, 196, 233, 281, 319] [33, 66, 85, 163, 197, 234, 282, 320] [34, 67, 86, 164, 198, 235, 283, 321] [35, 68, 87, 165, 199, 236, 284, 322] [36, 69, 88, 166, 200, 237, 285, 323] [37, 70, 89, 167, 201, 238, 286, 324] [38, 71, 90, 126, 202, 239, 287, 325] [39, 72, 91, 127, 203, 240, 288, 326] [40, 73, 92, 128, 204, 241, 289, 327] [41, 74, 93, 129, 205, 242, 290, 328] [0, 75, 94, 130, 206, 243, 291, 329] [1, 76, 95, 131, 207, 244, 292, 330] [2, 77, 96, 132, 208, 245, 293, 331] [3, 78, 97, 133, 209, 246, 252, 332] [4, 79, 98, 134, 168, 247, 253, 333] [5, 80, 99, 135, 169, 248, 254, 334] [6, 81, 100, 136, 170, 249, 255, 335] [7, 82, 101, 137, 171, 250, 256, 294] [8, 83, 102, 138, 172, 251, 257, 295] [13, 61, 84, 135, 170, 216, 252, 299] [14, 62, 85, 136, 171, 217, 253, 300] [15, 63, 86, 137, 172, 218, 254, 301] [16, 64, 87, 138, 173, 219, 255, 302] [17, 65, 88, 139, 174, 220, 256, 303] [18, 66, 89, 140, 175, 221, 257, 304] [19, 67, 90, 141, 176, 222, 258, 305] [20, 68, 91, 142, 177, 223, 259, 306] [21, 69, 92, 143, 178, 224, 260, 307] [22, 70, 93, 144, 179, 225, 261, 308] [23, 71, 94, 145, 180, 226, 262, 309] [24, 72, 95, 146, 181, 227, 263, 310] [25, 73, 96, 147, 182, 228, 264, 311] [26, 74, 97, 148, 183, 229, 265, 312] [27, 75, 98, 149, 184, 230, 266, 313] [28, 76, 99, 150, 185, 231, 267, 314] [29, 77, 100, 151, 186, 232, 268, 315] [30, 78, 101, 152, 187, 233, 269, 316] [31, 79, 102, 153, 188, 234, 270, 317] [32, 80, 103, 154, 189, 235, 271, 318] [33, 81, 104, 155, 190, 236, 272, 319] [34, 82, 105, 156, 191, 237, 273, 320] [35, 83, 106, 157, 192, 238, 274, 321] [36, 42, 107, 158, 193, 239, 275, 322] [37, 43, 108, 159, 194, 240, 276, 323] [38, 44, 109, 160, 195, 241, 277, 324] [39, 45, 110, 161, 196, 242, 278, 325] [40, 46, 111, 162, 197, 243, 279, 326] [41, 47, 112, 163, 198, 244, 280, 327] [0, 48, 113, 164, 199, 245, 281, 328] [1, 49, 114, 165, 200, 246, 282, 329] [2, 50, 115, 166, 201, 247, 283, 330] [3, 51, 116, 167, 202, 248, 284, 331] [4, 52, 117, 126, 203, 249, 285, 332] [5, 53, 118, 127, 204, 250, 286, 333] [6, 54, 119, 128, 205, 251, 287, 334] [7, 55, 120, 129, 206, 210, 288, 335] [8, 56, 121, 130, 207, 211, 289, 294] [9, 57, 122, 131, 208, 212, 290, 295] [10, 58, 123, 132, 209, 213, 291, 296] [11, 59, 124, 133, 168, 214, 292, 297] [12, 60, 125, 134, 169, 215, 293, 298] [19, 55, 93, 126, 174, 212, 257, 294] [20, 56, 94, 127, 175, 213, 258, 295] [21, 57, 95, 128, 176, 214, 259, 296] [22, 58, 96, 129, 177, 215, 260, 297] [23, 59, 97, 130, 178, 216, 261, 298] [24, 60, 98, 131, 179, 217, 262, 299] [25, 61, 99, 132, 180, 218, 263, 300] [26, 62, 100, 133, 181, 219, 264, 301] [27, 63, 101, 134, 182, 220, 265, 302] [28, 64, 102, 135, 183, 221, 266, 303] [29, 65, 103, 136, 184, 222, 267, 304] [30, 66, 104, 137, 185, 223, 268, 305] [31, 67, 105, 138, 186, 224, 269, 306] [32, 68, 106, 139, 187, 225, 270, 307] [33, 69, 107, 140, 188, 226, 271, 308] [34, 70, 108, 141, 189, 227, 272, 309] [35, 71, 109, 142, 190, 228, 273, 310] [36, 72, 110, 143, 191, 229, 274, 311] [37, 73, 111, 144, 192, 230, 275, 312] [38, 74, 112, 145, 193, 231, 276, 313] [39, 75, 113, 146, 194, 232, 277, 314] [40, 76, 114, 147, 195, 233, 278, 315] [41, 77, 115, 148, 196, 234, 279, 316] [0, 78, 116, 149, 197, 235, 280, 317] [1, 79, 117, 150, 198, 236, 281, 318] [2, 80, 118, 151, 199, 237, 282, 319] [3, 81, 119, 152, 200, 238, 283, 320] [4, 82, 120, 153, 201, 239, 284, 321] [5, 83, 121, 154, 202, 240, 285, 322] [6, 42, 122, 155, 203, 241, 286, 323] [7, 43, 123, 156, 204, 242, 287, 324] [8, 44, 124, 157, 205, 243, 288, 325] [9, 45, 125, 158, 206, 244, 289, 326] [10, 46, 84, 159, 207, 245, 290, 327] [11, 47, 85, 160, 208, 246, 291, 328] [12, 48, 86, 161, 209, 247, 292, 329] [13, 49, 87, 162, 168, 248, 293, 330] [14, 50, 88, 163, 169, 249, 252, 331] [15, 51, 89, 164, 170, 250, 253, 332] [16, 52, 90, 165, 171, 251, 254, 333] [17, 53, 91, 166, 172, 210, 255, 334] [18, 54, 92, 167, 173, 211, 256, 335]
Code ID 336-28-20 · download JSON · raw on GitHub