Covering designs
C(49,13,3) ≤ 81
The smallest known (49,13,3) covering design uses 81 blocks, one fewer than the 82 known before. Quantavane published it on 22 October 2026. The Schönheim bound shows that at least 76 blocks are needed, so C(49,13,3) lies between 76 and 81.
Uploaded to coveringrepository.com on 25 October 2026 as Arhan Canli (Quantavane), where it became the listed record.
The record
| Ours | 81 blocks |
|---|---|
| Previous best | 82 blocks |
| Previous source | the La Jolla Covering Repository's final data (Daniel M. Gordon, CC BY 4.0), a covering by Franco Atzeni (2024) |
| Published | 22 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 triple of the 49 points lies in at least one of the 81 blocks of 13 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(49,13,3)?
The fewest blocks of 13 points, drawn from 49 points, such that every triple of points lies in some block.
Is 81 the smallest possible?
Not known. The Schönheim bound gives at least 76; 81 is the smallest covering found so far.
What was the best known before?
82 blocks: the La Jolla Covering Repository's final data (Daniel M. Gordon, CC BY 4.0), a covering by Franco Atzeni (2024).