[[84,10,11]] · QEC Challenge

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[[84,10,11]] d ≤

kd²/n

14.405

X/Z

0.0105

6.0828

layers

swaps

478

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (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, 11 qubits)

[5, 9, 12, 33, 35, 42, 52, 54, 58, 80, 81]

d_Z 11 · witness weight 11 (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, 11 qubits)

[0, 13, 14, 25, 56, 59, 60, 69, 77, 78, 81]

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)×42 (2,5)×126 (3,4)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 42 (1,5): 42 (2,5): 126 (2,6): 294 (2,7): 588 (2,8): 336 (3,4): 42 (3,5): 252 (3,6): 462 (3,7): 2394 (3,8): 5292 (3,9): 7994 (3,10): 8358 (3,11): 3780 (3,12): 546 (3,13): 210

trapping sets H_Z (1,4)×42 (2,5)×126 (3,4)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 42 (1,5): 42 (2,5): 126 (2,6): 294 (2,7): 588 (2,8): 336 (3,4): 42 (3,5): 252 (3,6): 462 (3,7): 2394 (3,8): 5292 (3,9): 7994 (3,10): 8358 (3,11): 3780 (3,12): 546 (3,13): 210

witness diameter X 6.7082 · Z 5.831 (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 (44)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 478 nearest-neighbor SWAPs per round in total, at most 9 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 C42 = SmallGroup(42, 6) (order 42), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 84; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 3, 8, 41}, b = {0, 5, 19, 39} 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

[[84,10,11]] — two-block group-algebra code over C42

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| = 84

Parameters are also spelled out in provenance.construction of codes/84-10-11.json.

Parity checks

X-checks 42 (max weight 9) · Z-checks 42 (max weight 9)

H_X (42 checks, sparse supports)

[0, 6, 10, 26, 32, 42, 57, 72, 77] [1, 7, 9, 29, 35, 43, 54, 74, 75] [0, 2, 15, 31, 37, 44, 47, 62, 81] [0, 3, 12, 16, 32, 43, 45, 63, 78] [1, 4, 12, 34, 39, 45, 46, 65, 79] [1, 5, 13, 15, 35, 42, 47, 60, 80] [2, 5, 6, 36, 40, 48, 52, 67, 83] [2, 3, 7, 21, 37, 46, 49, 53, 68] [0, 3, 8, 18, 22, 47, 50, 69, 82] [3, 4, 9, 38, 41, 49, 51, 70, 82] [4, 5, 10, 18, 39, 44, 50, 52, 71] [1, 5, 11, 19, 21, 45, 53, 66, 83] [6, 7, 11, 12, 40, 51, 54, 58, 73] [2, 7, 8, 13, 27, 52, 55, 59, 74] [3, 8, 14, 24, 28, 48, 53, 56, 75] [8, 9, 10, 15, 41, 48, 55, 57, 76] [4, 10, 11, 16, 24, 49, 56, 58, 77] [5, 11, 17, 25, 27, 50, 51, 59, 72] [6, 12, 13, 17, 18, 57, 60, 64, 79] [7, 13, 14, 19, 33, 42, 58, 61, 65] [8, 14, 20, 30, 34, 54, 59, 62, 80] [9, 14, 15, 16, 21, 54, 61, 63, 81] [10, 16, 17, 22, 30, 43, 55, 62, 64] [11, 17, 23, 31, 33, 56, 57, 65, 78] [12, 18, 19, 23, 24, 44, 63, 66, 70] [13, 19, 20, 25, 38, 45, 64, 67, 71] [14, 20, 26, 36, 39, 60, 65, 68, 83] [15, 20, 21, 22, 27, 46, 60, 67, 69] [16, 22, 23, 28, 36, 47, 61, 68, 70] [17, 23, 29, 37, 38, 62, 63, 71, 82] [18, 24, 25, 29, 30, 49, 69, 72, 76] [19, 25, 26, 31, 41, 50, 70, 73, 77] [4, 20, 26, 32, 40, 51, 66, 71, 74] [21, 26, 27, 28, 33, 52, 66, 73, 75] [22, 28, 29, 34, 40, 53, 67, 74, 76] [2, 23, 29, 35, 41, 48, 68, 69, 77] [24, 30, 31, 35, 36, 55, 75, 78, 81] [9, 25, 31, 32, 37, 43, 56, 76, 79] [27, 32, 33, 34, 38, 58, 72, 79, 80] [6, 28, 34, 35, 39, 42, 59, 73, 81] [1, 30, 36, 37, 40, 46, 61, 80, 82] [0, 33, 38, 39, 41, 44, 64, 78, 83]

H_Z (42 checks, sparse supports)

[0, 5, 19, 39, 42, 44, 45, 50, 83] [1, 3, 22, 37, 43, 46, 47, 53, 82] [2, 10, 24, 41, 44, 48, 49, 55, 77] [3, 4, 11, 25, 45, 49, 50, 51, 56] [4, 7, 27, 40, 46, 51, 52, 58, 74] [2, 5, 8, 28, 47, 48, 52, 53, 59] [6, 14, 15, 35, 42, 48, 54, 60, 81] [7, 9, 16, 30, 43, 49, 54, 55, 61] [8, 10, 17, 31, 50, 55, 56, 57, 62] [9, 12, 17, 32, 43, 51, 57, 63, 79] [6, 10, 13, 33, 42, 52, 57, 58, 64] [7, 11, 14, 34, 53, 54, 58, 59, 65] [1, 12, 20, 21, 45, 46, 54, 60, 66] [13, 15, 22, 36, 47, 55, 60, 61, 67] [14, 16, 23, 37, 56, 61, 62, 63, 68] [0, 15, 18, 23, 44, 47, 57, 63, 69] [12, 16, 19, 38, 45, 58, 63, 64, 70] [13, 17, 20, 39, 59, 60, 64, 65, 71] [5, 18, 26, 27, 50, 52, 60, 66, 72] [19, 21, 28, 40, 53, 61, 66, 67, 73] [2, 20, 22, 29, 62, 67, 68, 69, 74] [3, 21, 24, 29, 49, 53, 63, 69, 75] [18, 22, 25, 41, 50, 64, 69, 70, 76] [4, 19, 23, 26, 65, 66, 70, 71, 77] [11, 24, 32, 33, 56, 58, 66, 72, 78] [6, 25, 27, 34, 59, 67, 72, 73, 79] [7, 26, 28, 35, 42, 68, 73, 74, 75] [8, 27, 30, 35, 55, 59, 69, 75, 80] [9, 24, 28, 31, 56, 70, 75, 76, 81] [10, 25, 29, 32, 43, 71, 72, 76, 77] [0, 17, 30, 38, 62, 64, 72, 78, 82] [12, 31, 33, 39, 44, 65, 73, 78, 79] [1, 13, 32, 34, 42, 45, 74, 79, 80] [1, 14, 33, 36, 61, 65, 75, 80, 83] [15, 30, 34, 37, 46, 62, 76, 80, 81] [0, 16, 31, 35, 43, 47, 77, 78, 81] [3, 23, 36, 41, 48, 68, 70, 78, 82] [4, 18, 37, 38, 44, 49, 71, 79, 82] [5, 20, 38, 40, 51, 67, 71, 80, 83] [2, 21, 36, 39, 46, 52, 68, 81, 83] [8, 9, 29, 40, 48, 54, 74, 76, 82] [6, 11, 26, 41, 51, 57, 73, 77, 83]

Code ID 84-10-11 · download JSON · raw on GitHub