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[[120,12,14]] d ≤
n
120
k
12
d
14
kd²/n
19.6
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (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, 14 qubits)
[6, 21, 22, 26, 41, 43, 67, 79, 85, 86, 98, 108, 110, 119]
d_Z 14 · witness weight 14 (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, 14 qubits)
[5, 13, 15, 17, 20, 26, 29, 33, 49, 59, 60, 76, 84, 115]
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)×120 (2,6)×1680 (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): 120 (2,6): 1680 (3,6): 1680 (3,8): 30240 (3,10): 3360
trapping sets H_Z (1,4)×120 (2,6)×1680 (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): 120 (2,6): 1680 (3,6): 1680 (3,8): 30240 (3,10): 3360

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C30 x C2 = SmallGroup(60, 13) (order 60), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 120; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 34, 37}, b = {0, 4, 36, 53} 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

[[120,12,14]] — two-block group-algebra code over C30 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| = 120

Parameters are also spelled out in provenance.construction of codes/120-12-14.json.

Parity checks

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