← back to the board
[[120,16,10]] d ≤
n
120
k
16
d
10
kd²/n
13.333
w
7
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (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, 10 qubits)
[16, 27, 38, 48, 71, 81, 92, 103, 108, 115]
d_Z 10 · witness weight 10 (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, 10 qubits)
[6, 21, 38, 44, 46, 52, 63, 70, 84, 116]
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)×60 (2,4)×180 (3,3)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 60 (1,4): 60 (2,4): 180 (2,5): 720 (2,6): 360 (3,3): 60 (3,5): 780 (3,6): 6360 (3,7): 8700 (3,8): 3240 (3,9): 1080 (3,10): 240
trapping sets H_Z (1,3)×60 (2,4)×180 (3,3)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 60 (1,4): 60 (2,4): 180 (2,5): 720 (2,6): 360 (3,3): 60 (3,5): 780 (3,6): 6360 (3,7): 8700 (3,8): 3240 (3,9): 1080 (3,10): 240

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, 6, 25, 42}, b = {0, 19, 51} 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

[[120,16,10]] — 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/120-16-10.json.

Parity checks

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