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[[146,20,12]] d ≤
n
146
k
20
d
12
kd²/n
19.726
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (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, 12 qubits)
[11, 46, 54, 57, 81, 84, 86, 97, 113, 119, 130, 132]
d_Z 12 · witness weight 12 (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, 12 qubits)
[11, 27, 40, 46, 53, 56, 97, 105, 126, 132, 140, 143]
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)×146 (2,6)×2044 (3,6)×2336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 146 (2,6): 2044 (3,6): 2336 (3,8): 35916 (3,10): 4088
trapping sets H_Z (1,4)×146 (2,6)×2044 (3,6)×2336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 146 (2,6): 2044 (3,6): 2336 (3,8): 35916 (3,10): 4088

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, 3, 14, 60}, b = {0, 1, 44, 52} 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

[[146,20,12]] — 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/146-20-12.json.

Parity checks

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