Covering designs
C(41,9,2) ≤ 27
The smallest known (41,9,2) covering design uses 27 blocks, one fewer than the 28 known before. Quantavane published it on 23 October 2026. The Schönheim bound shows that at least 23 blocks are needed, so C(41,9,2) lies between 23 and 27.
Uploaded to coveringrepository.com on 25 October 2026 as Arhan Canli (Quantavane), where it became the listed record.
The record
| Ours | 27 blocks |
|---|---|
| Previous best | 28 blocks |
| Previous source | the La Jolla Covering Repository's final data (Daniel M. Gordon, CC BY 4.0), a covering by Colin Barker |
| Published | 23 October 2026 |
| Status | Uploaded to coveringrepository.com on 25 October 2026 as Arhan Canli (Quantavane), where it became the listed record. |
| Check | Checked exhaustively by three independent programs; verify.py in the public repository checks it again with plain Python. |
Every pair of the 41 points lies in at least one of the 27 blocks of 9 points.
How it was found and checked
Found by a compiled local search that alternates simulated annealing with breakout weighting. Checked exhaustively by three independent programs; verify.py in the public repository checks it again with plain Python.
git clone https://github.com/arhancanli/quantavane-covering-records
cd quantavane-covering-records
python3 verify.pyTake v points and blocks of k points each. A (v,k,t) covering design is a set of blocks such that every group of t points sits inside at least one block. C(v,k,t) is the fewest blocks that can do it. For most sizes no one knows the exact answer, so public tables keep the best covering anyone has found. Beating an entry means building a valid covering with fewer blocks.
Questions
What is C(41,9,2)?
The fewest blocks of 9 points, drawn from 41 points, such that every pair of points lies in some block.
Is 27 the smallest possible?
Not known. The Schönheim bound gives at least 23; 27 is the smallest covering found so far.
What was the best known before?
28 blocks: the La Jolla Covering Repository's final data (Daniel M. Gordon, CC BY 4.0), a covering by Colin Barker.