[[96,22,8]] · QEC Challenge

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[[96,22,8]] d ≤

kd²/n

14.667

X/Z

0.0092

6.3246

layers

swaps

593

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · w_X = 8, w_Z = 8 (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)

[4, 25, 27, 30, 54, 57, 73, 78]

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)

[14, 20, 40, 43, 50, 52, 59, 60]

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×96 (2,4)×96 (3,4)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 96 (2,4): 96 (2,6): 1152 (3,4): 96 (3,6): 3456 (3,8): 18432 (3,10): 1536

trapping sets H_Z (1,4)×96 (2,4)×96 (3,4)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 96 (2,4): 96 (2,6): 1152 (3,4): 96 (3,6): 3456 (3,8): 18432 (3,10): 1536

witness diameter X 5.3852 · 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 (49)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 593 nearest-neighbor SWAPs per round in total, at most 11 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 C3 x (C8 : C2) = SmallGroup(48, 24) (order 48), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 96; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 4, 7, 32}, b = {0, 4, 10, 30} as element indices. Check weight 8. 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 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

[[96,22,8]] — two-block group-algebra code over C3 x (C8 : C2)

Direction & hypothesis

Cell weight-8 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| = 96

Parameters are also spelled out in provenance.construction of codes/96-22-8.json.

Parity checks

X-checks 48 (max weight 8) · Z-checks 48 (max weight 8)

H_X (48 checks, sparse supports)

[0, 16, 31, 45, 48, 64, 72, 76] [1, 13, 23, 46, 49, 71, 79, 83] [2, 20, 27, 47, 50, 61, 75, 86] [3, 6, 23, 30, 50, 51, 52, 78] [0, 4, 20, 42, 48, 52, 77, 86] [4, 5, 35, 43, 52, 53, 87, 89] [6, 24, 34, 41, 54, 68, 82, 90] [0, 7, 27, 37, 54, 55, 56, 85] [1, 8, 24, 28, 49, 56, 84, 90] [8, 9, 29, 38, 56, 57, 91, 93] [1, 10, 34, 40, 48, 58, 59, 88] [2, 11, 31, 35, 50, 59, 76, 87] [11, 12, 36, 42, 59, 60, 77, 94] [13, 17, 37, 41, 58, 61, 62, 89] [1, 3, 14, 18, 51, 53, 59, 62] [8, 14, 15, 19, 60, 62, 63, 64] [5, 16, 36, 47, 53, 61, 64, 94] [2, 16, 17, 44, 49, 65, 66, 92] [6, 13, 18, 38, 54, 66, 83, 91] [18, 19, 28, 39, 66, 67, 84, 95] [3, 20, 40, 45, 65, 68, 69, 93] [2, 4, 7, 21, 55, 57, 66, 69] [5, 11, 21, 22, 67, 69, 70, 71] [9, 23, 39, 41, 57, 68, 71, 95] [7, 24, 44, 46, 51, 72, 73, 94] [6, 8, 10, 25, 52, 58, 60, 73] [9, 18, 25, 26, 53, 73, 74, 75] [12, 27, 43, 45, 60, 72, 75, 89] [7, 13, 28, 32, 61, 63, 73, 76] [21, 28, 29, 33, 74, 76, 77, 78] [9, 15, 30, 34, 48, 63, 75, 78] [10, 30, 31, 47, 55, 79, 80, 95] [0, 11, 17, 32, 56, 65, 67, 80] [4, 12, 32, 33, 57, 80, 81, 82] [19, 29, 34, 46, 67, 79, 82, 93] [10, 14, 20, 35, 68, 70, 80, 83] [15, 25, 35, 36, 81, 83, 84, 85] [12, 16, 22, 37, 49, 70, 82, 85] [17, 21, 24, 38, 62, 72, 74, 86] [22, 32, 38, 39, 63, 86, 87, 88] [19, 23, 26, 40, 50, 64, 74, 88] [22, 29, 41, 44, 51, 77, 88, 89] [3, 25, 31, 42, 69, 79, 81, 90] [14, 26, 42, 43, 70, 90, 91, 92] [5, 27, 33, 44, 54, 71, 81, 92] [26, 30, 36, 45, 55, 84, 92, 93] [33, 37, 39, 46, 58, 78, 87, 94] [15, 40, 43, 47, 65, 85, 91, 95]

H_Z (48 checks, sparse supports)

[0, 4, 10, 30, 48, 52, 55, 80] [1, 8, 17, 37, 49, 56, 58, 62] [2, 3, 11, 40, 50, 59, 65, 69] [3, 14, 24, 41, 51, 62, 68, 90] [3, 4, 5, 25, 52, 53, 69, 81] [5, 14, 16, 26, 53, 64, 70, 92] [6, 7, 18, 44, 51, 54, 66, 73] [7, 21, 31, 45, 55, 69, 72, 76] [7, 8, 9, 32, 56, 57, 63, 73] [9, 21, 23, 33, 57, 71, 74, 78] [10, 13, 25, 46, 58, 73, 79, 83] [10, 11, 12, 14, 59, 60, 70, 80] [12, 15, 25, 27, 60, 75, 81, 85] [2, 13, 16, 28, 49, 61, 66, 76] [13, 14, 15, 38, 62, 63, 83, 91] [15, 28, 30, 39, 63, 78, 84, 95] [0, 15, 16, 40, 48, 64, 65, 85] [17, 20, 32, 47, 61, 65, 80, 86] [17, 18, 19, 21, 62, 66, 67, 74] [19, 22, 32, 34, 63, 67, 82, 88] [6, 20, 23, 35, 50, 52, 68, 83] [20, 21, 22, 42, 69, 70, 77, 86] [22, 35, 37, 43, 70, 85, 87, 89] [1, 22, 23, 44, 49, 51, 71, 88] [0, 24, 27, 38, 54, 56, 72, 86] [24, 25, 26, 28, 73, 74, 84, 90] [26, 29, 38, 40, 74, 88, 91, 93] [2, 26, 27, 30, 50, 55, 75, 92] [0, 11, 28, 29, 56, 67, 76, 77] [4, 12, 29, 41, 57, 77, 82, 89] [3, 29, 30, 46, 51, 78, 79, 93] [1, 31, 34, 42, 48, 59, 79, 90] [31, 32, 33, 35, 76, 80, 81, 87] [33, 36, 42, 44, 77, 81, 92, 94] [6, 33, 34, 37, 54, 58, 78, 82] [1, 18, 35, 36, 53, 59, 83, 84] [8, 19, 36, 45, 60, 64, 84, 93] [7, 36, 37, 47, 55, 61, 85, 94] [2, 4, 38, 39, 57, 66, 86, 87] [5, 11, 39, 46, 67, 71, 87, 94] [10, 39, 40, 41, 58, 68, 88, 95] [5, 13, 27, 41, 54, 61, 71, 89] [6, 8, 42, 43, 52, 60, 90, 91] [9, 18, 43, 47, 53, 75, 91, 95] [17, 43, 44, 45, 65, 72, 89, 92] [9, 20, 34, 45, 48, 68, 75, 93] [12, 16, 24, 46, 49, 72, 82, 94] [19, 23, 31, 47, 50, 64, 79, 95]

Code ID 96-22-8 · download JSON · raw on GitHub