[[288,32,18]] · QEC Challenge

[[288,32,18]] d ≤

288

kd²/n

36.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 9, w_Z = 9 (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)

[149, 150, 158, 165, 166, 171, 185, 187, 195, 196, 204, 205, 223, 225, 242, 243, 259, 271]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[147, 162, 168, 174, 188, 195, 212, 213, 214, 233, 234, 240, 242, 246, 262, 266, 271, 274]

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 9 · H_Z 9

qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)

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 (1,6): 144 (2,4): 432 (2,7): 2592 (2,10): 2160 (3,3): 144 (3,5): 1296 (3,6): 576 (3,7): 144 (3,8): 28080 (3,9): 2880 (3,10): 2592 (3,11): 71712 (3,12): 3888 (3,13): 6480 (3,14): 42336 (3,16): 2880

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

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Two-block group-algebra code C(a,b) over SmallGroup(144, 101)=C12 x C12 from arXiv:2610.13880 (Gideon Lee), database entry da0e585965415813. a=[0, 2, 24, 35, 58, 93], b=[0, 55, 142].

model Beam 1.0 (claimed, not verified)

builds on arXiv:2610.13880

date 2026-10-21

notes Imported from https://github.com/gideonlsx/bicycle-race entry da0e585965415813. Labels: ['front', 'front-2D'].

family generalized bicycle (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

[[288,32,18]] — two-block group-algebra code from arXiv:2610.13880

Direction & hypothesis

Unrestricted / weight-9plus cell. Imported from the public database of arXiv:2610.13880 (Gideon Lee, "Bicycle Race"), entry da0e585965415813.

What was searched

Reconstruction of a published two-block group-algebra code C(a,b) over SmallGroup(144, 101) = C12 x C12, with a=[0, 2, 24, 35, 58, 93], b=[0, 55, 142]. No original search performed here; parameters and blocks taken from the public database.

Evidence trail

Distance is a witness-backed upper bound: d_X <= 18, d_Z <= 18, both with witness logicals of weight 18. Verified by qldpc_verify (CSS commutation, k recomputation, witness validity) and RIS refutation (8000 trials, no lighter logical found). Confidence: upper_bound.

Dead ends

None recorded; this is a reproduction of a published code.

Tools

Imported via research/kit/submit.py with --reuse-witnesses; model Beam 1.0.

Reproduction

Group: SmallGroup(144, 101). a = [0, 2, 24, 35, 58, 93]. b = [0, 55, 142]. Rebuild with research/kit/rebuild.py or scripts/codes.py from github.com/gideonlsx/bicycle-race.

Parity checks

X-checks 144 (max weight 9) · Z-checks 144 (max weight 9)

H_X (144 checks, sparse supports)

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

H_Z (144 checks, sparse supports)

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

Code ID 288-32-18 · download JSON · raw on GitHub