[[66,44,2]] · QEC Challenge

[[66,44,2]] d =

kd²/n

2.667

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 2 · witness weight 2 (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, 2 qubits)

[29, 32]

d_Z 2 · witness weight 2 (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, 2 qubits)

[50, 54]

certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT

X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 9 · H_Z 9

qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)

trapping sets H_X (1,3)×33 (2,0)×66 (3,0)×297 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 33 (1,6): 33 (2,0): 66 (2,3): 198 (2,6): 99 (3,0): 297 (3,3): 803 (3,6): 704 (3,9): 297

trapping sets H_Z (1,3)×33 (2,0)×66 (3,0)×297 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 33 (1,6): 33 (2,0): 66 (2,3): 198 (2,6): 99 (3,0): 297 (3,3): 803 (3,6): 704 (3,9): 297

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C33 = SmallGroup(33, 1) (order 33), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 66; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 3, 4, 6}, b = {0, 1, 3} as element indices. Check weight 9. 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

[[66,44,2]] — two-block group-algebra code over C33

Direction & hypothesis

Cell weight-9plus 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| = 66

Parameters are also spelled out in provenance.construction of codes/66-44-2.json.

Parity checks

X-checks 33 (max weight 9) · Z-checks 33 (max weight 9)

H_X (33 checks, sparse supports)

[0, 1, 3, 29, 31, 32, 33, 34, 36] [0, 1, 3, 29, 31, 32, 33, 34, 36] [0, 1, 2, 3, 4, 6, 35, 37, 39] [0, 1, 3, 29, 31, 32, 33, 34, 36] [0, 1, 2, 3, 4, 6, 35, 37, 39] [2, 4, 5, 6, 7, 9, 38, 40, 42] [0, 1, 2, 3, 4, 6, 35, 37, 39] [2, 4, 5, 6, 7, 9, 38, 40, 42] [5, 7, 8, 9, 10, 12, 41, 43, 45] [2, 4, 5, 6, 7, 9, 38, 40, 42] [5, 7, 8, 9, 10, 12, 41, 43, 45] [8, 10, 11, 12, 13, 15, 44, 46, 48] [5, 7, 8, 9, 10, 12, 41, 43, 45] [8, 10, 11, 12, 13, 15, 44, 46, 48] [11, 13, 14, 15, 16, 18, 47, 49, 51] [8, 10, 11, 12, 13, 15, 44, 46, 48] [11, 13, 14, 15, 16, 18, 47, 49, 51] [14, 16, 17, 18, 19, 21, 50, 52, 54] [11, 13, 14, 15, 16, 18, 47, 49, 51] [14, 16, 17, 18, 19, 21, 50, 52, 54] [17, 19, 20, 21, 22, 24, 53, 55, 57] [14, 16, 17, 18, 19, 21, 50, 52, 54] [17, 19, 20, 21, 22, 24, 53, 55, 57] [20, 22, 23, 24, 25, 27, 56, 58, 60] [17, 19, 20, 21, 22, 24, 53, 55, 57] [20, 22, 23, 24, 25, 27, 56, 58, 60] [23, 25, 26, 27, 28, 30, 59, 61, 63] [20, 22, 23, 24, 25, 27, 56, 58, 60] [23, 25, 26, 27, 28, 30, 59, 61, 63] [26, 28, 29, 30, 31, 32, 62, 64, 65] [23, 25, 26, 27, 28, 30, 59, 61, 63] [26, 28, 29, 30, 31, 32, 62, 64, 65] [26, 28, 29, 30, 31, 32, 62, 64, 65]

H_Z (33 checks, sparse supports)

[0, 1, 3, 33, 34, 35, 36, 37, 39] [0, 1, 3, 33, 34, 35, 36, 37, 39] [2, 4, 6, 35, 37, 38, 39, 40, 42] [0, 1, 3, 33, 34, 35, 36, 37, 39] [2, 4, 6, 35, 37, 38, 39, 40, 42] [5, 7, 9, 38, 40, 41, 42, 43, 45] [2, 4, 6, 35, 37, 38, 39, 40, 42] [5, 7, 9, 38, 40, 41, 42, 43, 45] [8, 10, 12, 41, 43, 44, 45, 46, 48] [5, 7, 9, 38, 40, 41, 42, 43, 45] [8, 10, 12, 41, 43, 44, 45, 46, 48] [11, 13, 15, 44, 46, 47, 48, 49, 51] [8, 10, 12, 41, 43, 44, 45, 46, 48] [11, 13, 15, 44, 46, 47, 48, 49, 51] [14, 16, 18, 47, 49, 50, 51, 52, 54] [11, 13, 15, 44, 46, 47, 48, 49, 51] [14, 16, 18, 47, 49, 50, 51, 52, 54] [17, 19, 21, 50, 52, 53, 54, 55, 57] [14, 16, 18, 47, 49, 50, 51, 52, 54] [17, 19, 21, 50, 52, 53, 54, 55, 57] [20, 22, 24, 53, 55, 56, 57, 58, 60] [17, 19, 21, 50, 52, 53, 54, 55, 57] [20, 22, 24, 53, 55, 56, 57, 58, 60] [23, 25, 27, 56, 58, 59, 60, 61, 63] [20, 22, 24, 53, 55, 56, 57, 58, 60] [23, 25, 27, 56, 58, 59, 60, 61, 63] [26, 28, 30, 59, 61, 62, 63, 64, 65] [23, 25, 27, 56, 58, 59, 60, 61, 63] [26, 28, 30, 59, 61, 62, 63, 64, 65] [29, 31, 32, 33, 34, 36, 62, 64, 65] [26, 28, 30, 59, 61, 62, 63, 64, 65] [29, 31, 32, 33, 34, 36, 62, 64, 65] [29, 31, 32, 33, 34, 36, 62, 64, 65]

Code ID 66-44-2 · download JSON · raw on GitHub