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[[146,18,9]] d ≤
n
146
k
18
d
9
kd²/n
9.986
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (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, 9 qubits)
[7, 8, 29, 37, 50, 59, 90, 112, 133]
d_Z 9 · witness weight 9 (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, 9 qubits)
[21, 51, 72, 74, 82, 103, 104, 125, 134]
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)×73 (2,4)×219 (3,3)×73 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 73 (1,4): 73 (2,4): 219 (2,5): 876 (2,6): 438 (3,3): 73 (3,5): 657 (3,6): 7884 (3,7): 11461 (3,8): 3504 (3,9): 1314 (3,10): 292
trapping sets H_Z (1,3)×73 (2,4)×219 (3,3)×73 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 73 (1,4): 73 (2,4): 219 (2,5): 876 (2,6): 438 (3,3): 73 (3,5): 657 (3,6): 7884 (3,7): 11461 (3,8): 3504 (3,9): 1314 (3,10): 292

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C73 = SmallGroup(73, 1) (order 73), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 146; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 10, 66}, b = {0, 13, 42} 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

[[146,18,9]] — two-block group-algebra code over C73

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| = 146

Parameters are also spelled out in provenance.construction of codes/146-18-9.json.

Parity checks

X-checks 73 (max weight 7) · Z-checks 73 (max weight 7)
H_X (73 checks, sparse supports)
[0, 7, 63, 72, 73, 104, 133] [0, 1, 8, 64, 74, 105, 134] [1, 2, 9, 65, 75, 106, 135] [2, 3, 10, 66, 76, 107, 136] [3, 4, 11, 67, 77, 108, 137] [4, 5, 12, 68, 78, 109, 138] [5, 6, 13, 69, 79, 110, 139] [6, 7, 14, 70, 80, 111, 140] [7, 8, 15, 71, 81, 112, 141] [8, 9, 16, 72, 82, 113, 142] [0, 9, 10, 17, 83, 114, 143] [1, 10, 11, 18, 84, 115, 144] [2, 11, 12, 19, 85, 116, 145] [3, 12, 13, 20, 73, 86, 117] [4, 13, 14, 21, 74, 87, 118] [5, 14, 15, 22, 75, 88, 119] [6, 15, 16, 23, 76, 89, 120] [7, 16, 17, 24, 77, 90, 121] [8, 17, 18, 25, 78, 91, 122] [9, 18, 19, 26, 79, 92, 123] [10, 19, 20, 27, 80, 93, 124] [11, 20, 21, 28, 81, 94, 125] [12, 21, 22, 29, 82, 95, 126] [13, 22, 23, 30, 83, 96, 127] [14, 23, 24, 31, 84, 97, 128] [15, 24, 25, 32, 85, 98, 129] [16, 25, 26, 33, 86, 99, 130] [17, 26, 27, 34, 87, 100, 131] [18, 27, 28, 35, 88, 101, 132] [19, 28, 29, 36, 89, 102, 133] [20, 29, 30, 37, 90, 103, 134] [21, 30, 31, 38, 91, 104, 135] [22, 31, 32, 39, 92, 105, 136] [23, 32, 33, 40, 93, 106, 137] [24, 33, 34, 41, 94, 107, 138] [25, 34, 35, 42, 95, 108, 139] [26, 35, 36, 43, 96, 109, 140] [27, 36, 37, 44, 97, 110, 141] [28, 37, 38, 45, 98, 111, 142] [29, 38, 39, 46, 99, 112, 143] [30, 39, 40, 47, 100, 113, 144] [31, 40, 41, 48, 101, 114, 145] [32, 41, 42, 49, 73, 102, 115] [33, 42, 43, 50, 74, 103, 116] [34, 43, 44, 51, 75, 104, 117] [35, 44, 45, 52, 76, 105, 118] [36, 45, 46, 53, 77, 106, 119] [37, 46, 47, 54, 78, 107, 120] [38, 47, 48, 55, 79, 108, 121] [39, 48, 49, 56, 80, 109, 122] [40, 49, 50, 57, 81, 110, 123] [41, 50, 51, 58, 82, 111, 124] [42, 51, 52, 59, 83, 112, 125] [43, 52, 53, 60, 84, 113, 126] [44, 53, 54, 61, 85, 114, 127] [45, 54, 55, 62, 86, 115, 128] [46, 55, 56, 63, 87, 116, 129] [47, 56, 57, 64, 88, 117, 130] [48, 57, 58, 65, 89, 118, 131] [49, 58, 59, 66, 90, 119, 132] [50, 59, 60, 67, 91, 120, 133] [51, 60, 61, 68, 92, 121, 134] [52, 61, 62, 69, 93, 122, 135] [53, 62, 63, 70, 94, 123, 136] [54, 63, 64, 71, 95, 124, 137] [55, 64, 65, 72, 96, 125, 138] [0, 56, 65, 66, 97, 126, 139] [1, 57, 66, 67, 98, 127, 140] [2, 58, 67, 68, 99, 128, 141] [3, 59, 68, 69, 100, 129, 142] [4, 60, 69, 70, 101, 130, 143] [5, 61, 70, 71, 102, 131, 144] [6, 62, 71, 72, 103, 132, 145]
H_Z (73 checks, sparse supports)
[0, 13, 42, 73, 74, 83, 139] [1, 14, 43, 74, 75, 84, 140] [2, 15, 44, 75, 76, 85, 141] [3, 16, 45, 76, 77, 86, 142] [4, 17, 46, 77, 78, 87, 143] [5, 18, 47, 78, 79, 88, 144] [6, 19, 48, 79, 80, 89, 145] [7, 20, 49, 73, 80, 81, 90] [8, 21, 50, 74, 81, 82, 91] [9, 22, 51, 75, 82, 83, 92] [10, 23, 52, 76, 83, 84, 93] [11, 24, 53, 77, 84, 85, 94] [12, 25, 54, 78, 85, 86, 95] [13, 26, 55, 79, 86, 87, 96] [14, 27, 56, 80, 87, 88, 97] [15, 28, 57, 81, 88, 89, 98] [16, 29, 58, 82, 89, 90, 99] [17, 30, 59, 83, 90, 91, 100] [18, 31, 60, 84, 91, 92, 101] [19, 32, 61, 85, 92, 93, 102] [20, 33, 62, 86, 93, 94, 103] [21, 34, 63, 87, 94, 95, 104] [22, 35, 64, 88, 95, 96, 105] [23, 36, 65, 89, 96, 97, 106] [24, 37, 66, 90, 97, 98, 107] [25, 38, 67, 91, 98, 99, 108] [26, 39, 68, 92, 99, 100, 109] [27, 40, 69, 93, 100, 101, 110] [28, 41, 70, 94, 101, 102, 111] [29, 42, 71, 95, 102, 103, 112] [30, 43, 72, 96, 103, 104, 113] [0, 31, 44, 97, 104, 105, 114] [1, 32, 45, 98, 105, 106, 115] [2, 33, 46, 99, 106, 107, 116] [3, 34, 47, 100, 107, 108, 117] [4, 35, 48, 101, 108, 109, 118] [5, 36, 49, 102, 109, 110, 119] [6, 37, 50, 103, 110, 111, 120] [7, 38, 51, 104, 111, 112, 121] [8, 39, 52, 105, 112, 113, 122] [9, 40, 53, 106, 113, 114, 123] [10, 41, 54, 107, 114, 115, 124] [11, 42, 55, 108, 115, 116, 125] [12, 43, 56, 109, 116, 117, 126] [13, 44, 57, 110, 117, 118, 127] [14, 45, 58, 111, 118, 119, 128] [15, 46, 59, 112, 119, 120, 129] [16, 47, 60, 113, 120, 121, 130] [17, 48, 61, 114, 121, 122, 131] [18, 49, 62, 115, 122, 123, 132] [19, 50, 63, 116, 123, 124, 133] [20, 51, 64, 117, 124, 125, 134] [21, 52, 65, 118, 125, 126, 135] [22, 53, 66, 119, 126, 127, 136] [23, 54, 67, 120, 127, 128, 137] [24, 55, 68, 121, 128, 129, 138] [25, 56, 69, 122, 129, 130, 139] [26, 57, 70, 123, 130, 131, 140] [27, 58, 71, 124, 131, 132, 141] [28, 59, 72, 125, 132, 133, 142] [0, 29, 60, 126, 133, 134, 143] [1, 30, 61, 127, 134, 135, 144] [2, 31, 62, 128, 135, 136, 145] [3, 32, 63, 73, 129, 136, 137] [4, 33, 64, 74, 130, 137, 138] [5, 34, 65, 75, 131, 138, 139] [6, 35, 66, 76, 132, 139, 140] [7, 36, 67, 77, 133, 140, 141] [8, 37, 68, 78, 134, 141, 142] [9, 38, 69, 79, 135, 142, 143] [10, 39, 70, 80, 136, 143, 144] [11, 40, 71, 81, 137, 144, 145] [12, 41, 72, 73, 82, 138, 145]
Code ID 146-18-9 · download JSON · raw on GitHub