← back to the board
[[24,4,5]] d =
n
24
k
4
d
5
kd²/n
4.167
w
8
X/Z
1
g
0.0417
r
3.1623
layers
2
swaps
30

Share this result

Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (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, 5 qubits)
[0, 1, 4, 11, 13]
d_Z 5 · witness weight 5 (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, 5 qubits)
[3, 5, 10, 20, 22]
certificate exact, d = 5 · CryptoMiniSat 5.14.7 SAT
X: no logical < 5 exists; Z: no logical < 5 exists

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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×12 (2,2)×12 (3,2)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 12 (1,6): 12 (2,2): 12 (2,4): 48 (2,6): 96 (2,8): 30 (3,2): 36 (3,4): 456 (3,6): 612 (3,8): 312 (3,10): 24 (3,12): 4
trapping sets H_Z (1,2)×12 (2,2)×12 (3,2)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 12 (1,6): 12 (2,2): 12 (2,4): 48 (2,6): 96 (2,8): 30 (3,2): 36 (3,4): 456 (3,6): 612 (3,8): 312 (3,10): 24 (3,12): 4
witness diameter X 2.8284 · Z 3.1623 (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
r = 3.162
X checkZ checkqubit site (12)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 30 nearest-neighbor SWAPs per round in total, at most 3 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 D12 = SmallGroup(12, 4) (order 12), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 24; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 3, 7, 8}, b = {0, 6} 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

[[24,4,5]] — two-block group-algebra code over D12

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| = 24

Parameters are also spelled out in provenance.construction of codes/24-4-5.json.

Parity checks

X-checks 12 (max weight 8) · Z-checks 12 (max weight 8)
H_X (12 checks, sparse supports)
[0, 1, 2, 3, 7, 8, 12, 22] [0, 1, 4, 5, 9, 10, 13, 23] [0, 2, 4, 5, 6, 10, 14, 19] [0, 3, 5, 6, 7, 11, 14, 15] [1, 2, 4, 7, 8, 11, 16, 21] [1, 2, 3, 5, 8, 9, 16, 17] [2, 3, 6, 8, 9, 10, 12, 18] [0, 3, 4, 7, 9, 10, 18, 19] [0, 4, 5, 6, 8, 11, 13, 20] [1, 5, 6, 7, 9, 11, 20, 21] [1, 2, 6, 7, 10, 11, 15, 22] [3, 4, 8, 9, 10, 11, 17, 23]
H_Z (12 checks, sparse supports)
[0, 6, 12, 13, 14, 15, 19, 20] [1, 8, 12, 13, 16, 17, 21, 22] [2, 3, 12, 14, 16, 17, 18, 22] [3, 10, 12, 15, 17, 18, 19, 23] [4, 5, 13, 14, 16, 19, 20, 23] [5, 11, 13, 14, 15, 17, 20, 21] [6, 7, 14, 15, 18, 20, 21, 22] [2, 7, 12, 15, 16, 19, 21, 22] [8, 9, 12, 16, 17, 18, 20, 23] [4, 9, 13, 17, 18, 19, 21, 23] [0, 10, 13, 14, 18, 19, 22, 23] [1, 11, 15, 16, 20, 21, 22, 23]
Code ID 24-4-5 · download JSON · raw on GitHub