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[[240,28,16]] d ≤
n
240
k
28
d
16
kd²/n
29.867
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 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)
[4, 10, 20, 53, 69, 70, 84, 87, 112, 128, 142, 146, 159, 174, 189, 215]
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)
[2, 21, 22, 42, 53, 56, 73, 102, 121, 173, 179, 191, 193, 203, 204, 219]
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 4 · H_Z 4
trapping sets H_X (1,4)×240 (2,6)×3360 (3,6)×1680 (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 (2,6): 3360 (3,6): 1680 (3,8): 65520 (3,10): 6720
trapping sets H_Z (1,4)×240 (2,6)×3360 (3,6)×1680 (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 (2,6): 3360 (3,6): 1680 (3,8): 65520 (3,10): 6720

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, 29, 68, 116}, b = {0, 19, 95, 107} 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

[[240,28,16]] — 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/240-28-16.json.

Parity checks

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