[[90,20,8]] · QEC Challenge

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[[90,20,8]] d ≤

kd²/n

14.222

X/Z

0.0085

6.4031

layers

swaps

562

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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)

[27, 30, 38, 44, 60, 63, 78, 89]

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)

[1, 4, 20, 29, 45, 54, 65, 71]

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)×45 (2,6)×360 (3,6)×645 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 45 (1,5): 45 (2,6): 360 (2,7): 900 (2,8): 270 (3,6): 645 (3,7): 1575 (3,8): 4410 (3,9): 12780 (3,10): 11925 (3,11): 3465 (3,12): 1080 (3,13): 90

trapping sets H_Z (1,4)×45 (2,6)×360 (3,6)×645 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 45 (1,5): 45 (2,6): 360 (2,7): 900 (2,8): 270 (3,6): 645 (3,7): 1575 (3,8): 4410 (3,9): 12780 (3,10): 11925 (3,11): 3465 (3,12): 1080 (3,13): 90

witness diameter X 6.0 · Z 6.3246 (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 (45)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 562 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 C15 x C3 = SmallGroup(45, 2) (order 45), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 90; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 11, 37}, b = {0, 3, 23, 36} 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

[[90,20,8]] — two-block group-algebra code over C15 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| = 90

Parameters are also spelled out in provenance.construction of codes/90-20-8.json.

Parity checks

X-checks 45 (max weight 9) · Z-checks 45 (max weight 9)

H_X (45 checks, sparse supports)

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

H_Z (45 checks, sparse supports)

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

Code ID 90-20-8 · download JSON · raw on GitHub