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[[288,24,18]] d ≤
n
288
k
24
d
18
kd²/n
27.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[2, 5, 8, 14, 20, 29, 32, 35, 44, 50, 53, 62, 83, 104, 116, 125, 128, 140]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[149, 158, 161, 179, 182, 188, 191, 197, 203, 206, 209, 215, 227, 230, 239, 251, 263, 284]
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 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×144 (2,4)×432 (3,3)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 144 (1,5): 144 (2,4): 432 (2,6): 2160 (2,8): 1440 (3,3): 144 (3,5): 1296 (3,7): 22896 (3,9): 46368 (3,11): 20448 (3,13): 1440
trapping sets H_Z (1,3)×144 (2,4)×432 (3,3)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 144 (1,5): 144 (2,4): 432 (2,6): 2160 (2,8): 1440 (3,3): 144 (3,5): 1296 (3,7): 22896 (3,9): 46368 (3,11): 20448 (3,13): 1440

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Normal-subgroup lifted product over Z_12 x Z_12 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

[[288,24,18]] — 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: reduce the paper's normal-subgroup construction to the abelian group Z_12 x Z_12 and rebuild its supports.

Evidence trail

The matrices recompute to n=288, k=24, max check weight 8, and CSS commutation. The paper reports d=18. Pure-Python RIS found no lighter logical 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_12 x Z_12, use A={(0,2),(0,7),(1,1),(3,0),(11,11)} and B={(0,3),(1,0),(2,0)}, from the Supplementary Information of arXiv:2608.08996v1.

Parity checks

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