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[[294,24,16]] d ≤
n
294
k
24
d
16
kd²/n
20.898
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (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, 16 qubits)
[51, 56, 73, 75, 90, 92, 122, 124, 166, 198, 212, 222, 239, 263, 271, 285]
d_Z 16 · witness weight 16 (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, 16 qubits)
[3, 6, 22, 38, 81, 113, 142, 145, 149, 173, 175, 190, 205, 227, 246, 272]
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)×147 (2,4)×441 (3,3)×147 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 147 (1,4): 147 (2,4): 441 (2,5): 1764 (2,6): 882 (3,3): 147 (3,5): 1323 (3,6): 15239 (3,7): 23079 (3,8): 8967 (3,9): 2646 (3,10): 588
trapping sets H_Z (1,3)×147 (2,4)×441 (3,3)×147 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 147 (1,4): 147 (2,4): 441 (2,5): 1764 (2,6): 882 (3,3): 147 (3,5): 1323 (3,6): 15239 (3,7): 23079 (3,8): 8967 (3,9): 2646 (3,10): 588

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over (C7 x C7) : C3 = SmallGroup(147, 5) (order 147), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 294; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 8, 33}, b = {0, 27, 29} 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 2bga-nonabelian (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

[[294,24,16]] — two-block group-algebra code over (C7 x C7) : C3

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| = 294

Parameters are also spelled out in provenance.construction of codes/294-24-16.json.

Parity checks

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