[[62,10,9]] · QEC Challenge

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[[62,10,9]] d ≤

kd²/n

13.065

X/Z

0.0193

5.099

layers

swaps

262

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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)

[1, 5, 19, 27, 30, 38, 40, 48, 50]

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)

[0, 4, 20, 30, 34, 42, 45, 58, 61]

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)×31 (2,5)×93 (3,4)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 31 (1,5): 31 (2,5): 93 (2,6): 248 (2,7): 434 (2,8): 186 (3,4): 62 (3,5): 310 (3,6): 620 (3,7): 1860 (3,8): 3627 (3,9): 5518 (3,10): 4712 (3,11): 1364 (3,12): 279 (3,13): 62

trapping sets H_Z (1,4)×31 (2,5)×93 (3,4)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 31 (1,5): 31 (2,5): 93 (2,6): 248 (2,7): 434 (2,8): 186 (3,4): 62 (3,5): 310 (3,6): 620 (3,7): 1860 (3,8): 3627 (3,9): 5518 (3,10): 4712 (3,11): 1364 (3,12): 279 (3,13): 62

witness diameter X 5.0 · 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 (32)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 262 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 C31 = SmallGroup(31, 1) (order 31), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 62; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 3, 12, 13}, b = {0, 2, 18, 24} 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

[[62,10,9]] — two-block group-algebra code over C31

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| = 62

Parameters are also spelled out in provenance.construction of codes/62-10-9.json.

Parity checks

X-checks 31 (max weight 9) · Z-checks 31 (max weight 9)

H_X (31 checks, sparse supports)

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

H_Z (31 checks, sparse supports)

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

Code ID 62-10-9 · download JSON · raw on GitHub