[[60,8,9]] · QEC Challenge

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[[60,8,9]] d ≤

kd²/n

10.8

X/Z

0.016

5.099

layers

swaps

251

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (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, 9 qubits)

[0, 1, 23, 30, 40, 50, 52, 55, 56]

d_Z 9 · witness weight 9 (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, 9 qubits)

[6, 11, 12, 20, 32, 46, 49, 50, 56]

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 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 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)

trapping sets H_X (1,4)×30 (2,5)×120 (3,3)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 30 (1,5): 30 (2,5): 120 (2,6): 270 (2,7): 360 (2,8): 120 (3,3): 10 (3,5): 330 (3,6): 840 (3,7): 2130 (3,8): 3810 (3,9): 4530 (3,10): 3180 (3,11): 750 (3,12): 150

trapping sets H_Z (1,4)×30 (2,5)×120 (3,3)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 30 (1,5): 30 (2,5): 120 (2,6): 270 (2,7): 360 (2,8): 120 (3,3): 10 (3,5): 330 (3,6): 840 (3,7): 2130 (3,8): 3810 (3,9): 4530 (3,10): 3180 (3,11): 750 (3,12): 150

witness diameter X 4.4721 · Z 5.099 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

layout contributed by @MathysRennela · Simulated annealing over integer-grid site assignments with the repo's own research/local2d/fold_layout.search_layout, single layer at radius 4 then two layers at 7, plus a divisor scan of the two-block index convention Z_l x Z_m which gives the tighter embedding here; radius and per-site occupancy re-measured with the verifier's own rule. · 2026-10-13

X checkZ checkqubit site (31)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 251 nearest-neighbor SWAPs per round in total, at most 8 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C30 = SmallGroup(30, 4) (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, 2, 3, 7, 17}, b = {0, 3, 9, 13} 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 2D-local bilayer (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

[[60,8,9]] — two-block group-algebra code over C30

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/60-8-9.json.

Parity checks

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

H_X (30 checks, sparse supports)

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

H_Z (30 checks, sparse supports)

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

Code ID 60-8-9 · download JSON · raw on GitHub