[[60,16,6]] · QEC Challenge

[[60,16,6]] d =

kd²/n

9.6

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (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, 6 qubits)

[1, 15, 20, 24, 39, 58]

d_Z 6 · witness weight 6 (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, 6 qubits)

[8, 17, 33, 48, 53, 56]

certificate exact, d = 6 · CryptoMiniSat 5.14.7 SAT

X: no logical < 6 exists; Z: no logical < 6 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)×30 (2,4)×90 (3,3)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 30 (1,6): 30 (2,4): 90 (2,6): 30 (2,7): 540 (2,8): 135 (2,10): 90 (3,3): 30 (3,5): 390 (3,6): 390 (3,7): 1050 (3,8): 5550 (3,9): 2640 (3,10): 1140 (3,11): 4200 (3,12): 1030 (3,13): 270 (3,14): 210 (3,16): 30

trapping sets H_Z (1,3)×30 (2,4)×90 (3,3)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C3 x D10 = SmallGroup(30, 2) (order 30), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 60; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 5, 6, 22}, b = {0, 7, 25} 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 2bga-nonabelian (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

[[60,16,6]] — two-block group-algebra code over C3 x D10

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| = 60

Parameters are also spelled out in provenance.construction of codes/60-16-6.json.

Parity checks

X-checks 30 (max weight 9) · Z-checks 30 (max weight 9)

H_X (30 checks, sparse supports)

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

H_Z (30 checks, sparse supports)

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

Code ID 60-16-6 · download JSON · raw on GitHub