[[54,8,8]] · QEC Challenge

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[[54,8,8]] d ≤

kd²/n

9.481

X/Z

0.014

5.099

layers

swaps

214

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (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, 8 qubits)

[1, 2, 5, 7, 11, 23, 28, 39]

d_Z 8 · witness weight 8 (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, 8 qubits)

[4, 26, 36, 39, 42, 44, 48, 51]

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)×27 (2,5)×162 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 27 (1,5): 27 (2,5): 162 (2,6): 270 (2,7): 216 (2,8): 54 (3,3): 63 (3,4): 54 (3,5): 459 (3,6): 900 (3,7): 1917 (3,8): 3321 (3,9): 2646 (3,10): 1296 (3,11): 216 (3,12): 54

trapping sets H_Z (1,4)×27 (2,5)×162 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 27 (1,5): 27 (2,5): 162 (2,6): 270 (2,7): 216 (2,8): 54 (3,3): 63 (3,4): 54 (3,5): 459 (3,6): 900 (3,7): 1917 (3,8): 3321 (3,9): 2646 (3,10): 1296 (3,11): 216 (3,12): 54

witness diameter X 4.4721 · Z 4.4721 (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 (28)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 214 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 C9 x C3 = SmallGroup(27, 2) (order 27), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 54; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 4, 12}, b = {0, 1, 5, 18} 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

[[54,8,8]] — two-block group-algebra code over C9 x C3

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| = 54

Parameters are also spelled out in provenance.construction of codes/54-8-8.json.

Parity checks

X-checks 27 (max weight 9) · Z-checks 27 (max weight 9)

H_X (27 checks, sparse supports)

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

H_Z (27 checks, sparse supports)

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

Code ID 54-8-8 · download JSON · raw on GitHub