[[90,4,14]] · QEC Challenge

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[[90,4,14]] d ≤

kd²/n

8.711

X/Z

0.0054

6.3246

layers

swaps

619

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (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, 14 qubits)

[3, 4, 16, 26, 33, 37, 40, 42, 48, 51, 52, 58, 62, 86]

d_Z 14 · witness weight 14 (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, 14 qubits)

[10, 11, 17, 25, 28, 32, 45, 49, 51, 52, 54, 56, 67, 88]

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 2–7 (mean 4.5) · H_Z 2–7 (mean 4.5)

trapping sets H_X (1,2)×45 (2,2)×45 (3,2)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 45 (1,7): 45 (2,2): 45 (2,7): 630 (2,10): 135 (2,12): 675 (3,2): 45 (3,5): 45 (3,7): 4275 (3,8): 180 (3,9): 360 (3,10): 3735 (3,11): 135 (3,12): 14580 (3,13): 2250 (3,14): 1350 (3,15): 5415 (3,17): 4770 (3,19): 585

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

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

witness diameter X 7.2111 · Z 7.2111 (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 619 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 C45 = SmallGroup(45, 1) (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, 6, 23, 24, 42}, b = {0, 16} 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

[[90,4,14]] — two-block group-algebra code over C45

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/90-4-14.json.

Parity checks

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

H_X (45 checks, sparse supports)

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

H_Z (45 checks, sparse supports)

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

Code ID 90-4-14 · download JSON · raw on GitHub