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 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×35 (2,2)×35 (3,2)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 35
(1,3): 35
(2,2): 35
(2,3): 210
(2,4): 105
(3,2): 35
(3,3): 665
(3,4): 1155
(3,5): 420
(3,6): 210
(3,7): 35
trapping sets H_Z (1,2)×35 (2,2)×35 (3,2)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 35
(1,3): 35
(2,2): 35
(2,3): 210
(2,4): 105
(3,2): 35
(3,3): 665
(3,4): 1155
(3,5): 420
(3,6): 210
(3,7): 35
Construction & provenance
authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C35 = SmallGroup(35, 1) (order 35), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 70; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 13}, b = {0, 1} as element indices. Check weight 5. 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)
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 ≤ 6 (computed)
How this code was found
[[70,6,5]] — two-block group-algebra code over C35
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| = 70
Parameters are also spelled out in provenance.construction of codes/70-6-5.json.
Parity checks
X-checks 35 (max weight 5) · Z-checks 35 (max weight 5)
H_X (35 checks, sparse supports)
[0, 24, 29, 35, 45]
[1, 14, 28, 35, 36]
[0, 2, 32, 37, 50]
[3, 18, 31, 36, 38]
[1, 4, 19, 37, 39]
[2, 5, 34, 40, 55]
[6, 22, 33, 38, 41]
[3, 7, 23, 39, 42]
[4, 8, 24, 40, 43]
[5, 9, 10, 44, 60]
[10, 26, 34, 41, 45]
[6, 11, 27, 42, 46]
[7, 12, 28, 43, 47]
[0, 8, 13, 44, 48]
[9, 14, 15, 49, 64]
[10, 15, 30, 46, 50]
[11, 16, 31, 47, 51]
[1, 12, 17, 48, 52]
[2, 13, 18, 49, 53]
[14, 19, 20, 54, 67]
[15, 20, 33, 51, 55]
[3, 16, 21, 52, 56]
[4, 17, 22, 53, 57]
[5, 18, 23, 54, 58]
[19, 24, 25, 59, 69]
[6, 20, 25, 56, 60]
[7, 21, 26, 57, 61]
[8, 22, 27, 58, 62]
[9, 23, 28, 59, 63]
[11, 25, 29, 61, 64]
[12, 26, 30, 62, 65]
[13, 27, 31, 63, 66]
[16, 29, 32, 65, 67]
[17, 30, 33, 66, 68]
[21, 32, 34, 68, 69]
H_Z (35 checks, sparse supports)
[0, 1, 35, 37, 48]
[1, 3, 36, 39, 52]
[2, 4, 37, 40, 53]
[3, 6, 38, 42, 56]
[4, 7, 39, 43, 57]
[5, 8, 40, 44, 58]
[6, 10, 41, 46, 60]
[7, 11, 42, 47, 61]
[8, 12, 43, 48, 62]
[9, 13, 44, 49, 63]
[0, 10, 44, 45, 50]
[11, 15, 46, 51, 64]
[12, 16, 47, 52, 65]
[13, 17, 48, 53, 66]
[14, 18, 36, 49, 54]
[2, 15, 49, 50, 55]
[16, 20, 51, 56, 67]
[17, 21, 52, 57, 68]
[18, 22, 38, 53, 58]
[19, 23, 39, 54, 59]
[5, 20, 54, 55, 60]
[21, 25, 56, 61, 69]
[22, 26, 41, 57, 62]
[23, 27, 42, 58, 63]
[24, 28, 35, 43, 59]
[9, 25, 59, 60, 64]
[26, 29, 45, 61, 65]
[27, 30, 46, 62, 66]
[28, 31, 36, 47, 63]
[14, 29, 35, 64, 67]
[30, 32, 50, 65, 68]
[31, 33, 38, 51, 66]
[19, 32, 37, 67, 69]
[33, 34, 41, 55, 68]
[24, 34, 40, 45, 69]