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[[240,122,2]] d =
n
240
k
122
d
2
kd²/n
2.033
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (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, 2 qubits)
[154, 171]
d_Z 2 · witness weight 2 (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, 2 qubits)
[229, 236]
certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT
X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (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)×240 (2,0)×120 (3,4)×10080 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 240 (2,0): 120 (2,4): 1440 (3,4): 10080 (3,8): 1920
trapping sets H_Z (1,4)×240 (2,0)×120 (3,4)×10080 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 240 (2,0): 120 (2,4): 1440 (3,4): 10080 (3,8): 1920

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C60 x C2 = SmallGroup(120, 31) (order 120), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 240; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 6}, b = {0, 2, 31, 48} as element indices. Check weight 8. 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

[[240,122,2]] — two-block group-algebra code over C60 x C2

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| = 240

Parameters are also spelled out in provenance.construction of codes/240-122-2.json.

Parity checks

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