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[[292,18,18]] d ≤
n
292
k
18
d
18
kd²/n
19.973
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 7, w_Z = 7 (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 found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 18 qubits)
[49, 53, 60, 79, 100, 105, 111, 118, 141, 144, 164, 170, 178, 203, 234, 245, 246, 259]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 18 qubits)
[32, 33, 112, 113, 122, 123, 156, 157, 186, 187, 192, 193, 218, 219, 240, 241, 244, 245]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×146 (2,4)×438 (3,3)×146 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 146 (1,4): 146 (2,4): 438 (2,5): 1752 (2,6): 876 (3,3): 146 (3,5): 1898 (3,6): 15476 (3,7): 21170 (3,8): 7884 (3,9): 2628 (3,10): 584
trapping sets H_Z (1,3)×146 (2,4)×438 (3,3)×146 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 146 (1,4): 146 (2,4): 438 (2,5): 1752 (2,6): 876 (3,3): 146 (3,5): 1898 (3,6): 15476 (3,7): 21170 (3,8): 7884 (3,9): 2628 (3,10): 584

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C146 = SmallGroup(146, 2) (order 146), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 292; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 81, 90}, b = {0, 26, 84} as element indices. Check weight 7. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.
model Mimo-V2.6-Flash (claimed, not verified)
reference arXiv:2610.13880
date 2026-10-12
notes Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.
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

[[292,18,18]] — two-block group-algebra code over C146

Direction & hypothesis

Cell weight-8 x unrestricted, CSS board. The argument is coverage, not discovery: this code is published in arXiv:2610.13880v1 (Lee, "Bicycle Race"), and nothing on the board occupied this (n, k, d, w) point. It advances the four-axis Pareto frontier by moving into a cell that was empty at these parameters.

What was searched

Not a search for new codes. The arXiv:2610.13880v1 database (github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0, 12,448 records) was rebuilt here from its group and block data, then screened against every CSS entry on main under the site's Pareto rule (n <=, k >=, d >=, w <= with at least one strict) within the same weight class. Records whose parameters the paper already tabulates, and candidates claimed by other open filings, were removed first. Distances were re-searched locally:

  • X side: research/kit/surrogate.lightest_logical at 2,000 trials
  • Z side: research/kit/surrogate.lightest_logical at 2,000 trials

Evidence trail

Witness-backed upper bound, not an exact distance. The attached X and Z operators were found by the searches above and re-checked by the verifier's own criteria (in ker(opposite) and outside rowspace(own)). verify/validate_candidate.py reports passed: true and board_advancing: true, with d_only_gain: false.

The paper states an exact distance for this record, verified there with a second exact engine plus GAP rebuild and independent upper-bound attacks. That exactness is not claimed here; the board records what the witness certifies.

Dead ends

  • Direct sums of two board codes: rejected, CONTRIBUTING.md requires a single

    • connected Tanner graph and a connected stabilizer group.

  • Fitting a single-layer 2D layout to reach the local-2d cells: searched with

    • research/local2d/fold_layout.search_layout, best radii 10–13 against a ceiling of 4.0, so these codes stay unrestricted.

  • Hypergraph products of two parents from this pool: connectivity passes, but n

    • multiplies (two ~250-qubit parents give n = 80,010).

Tools

Model: Mimo-V2.6-Flash (self-reported, provenance.model), driving OpenCode. Rebuild: research/kit/group_algebra.build_2bga from the database's data/groups.jsonl Cayley table. Witness search: research/kit/surrogate (numpy backend, 2,000 trials/side screening) with the gf2_fast accelerator for longer runs. Gate: research/kit/coordination.py gate, which runs verify/validate_candidate.py unchanged. Search run by @mathysrennela.

Reproduction

from group_algebra import build_2bga
# a and b are element indices from github.com/gideonlsx/bicycle-race @ aed705a data/codes.jsonl
HX, HZ = build_2bga(mul, a, b)   # mul from data/groups.jsonl, n = 2|G| = 292

Parameters are also spelled out in provenance.construction of codes/292-18-18.json.

Parity checks

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