C-38

proof verified via machine

There is a (41,12,5) covering design with 2207 blocks, so C(41,12,5) ≤ 2207.

Machine-checked: this sentence is generated by the certificate checker from the attached certificate, and it is exactly what the check covers.

The author's statement — not covered by the machine check:

C(41,12,5) <= 2207. A 2207-block (41,12,5) covering design, beating the prior best of 2244 (coveringrepository.com) / 2245 (LJCR v1.2).

by caleb with Claude Opus 5.5 (Anthropic) · 2026-11-11 16:04 UTC · via web · hash 62362da628fe… · AI-assisted

Evidence

Explicit block file attached (one block per line, 1-indexed, sorted). Coverage verified exhaustively (0 uncovered t-subsets).

Prior best 2245 (LJCR v1.2), 2244 (coveringrepository.com).
Method: derived design from the 8,735-block (42,13,6) covering in LJCR Zenodo v1.2 (coverings/42.13.6.txt, credited to Jan de Heer and Steve Muir, 2026-02-28): take the blocks containing the minimum-degree point 16 (degree 2207; point degrees range 2207 to 3091), delete point 16, relabel {1..42}\{16} -> {1..41} order-preservingly. Anyone can reproduce the 2207 blocks from that public file with this rule.
All 749,398 5-subsets covered.
The 2207 blocks (74 kB) are too long to paste inline; they are in the attached data file 41.12.5.txt.
Author: Caleb Ashton.
SHA-256 of the block file: ec985f89dd793006ccf15346b6937419e1ce4e922034d73b18e769fb958778aa.
LJCR entry: https://ljcr.dmgordon.org/cover/show_cover.php?v=41&k=12&t=5

Citations

https://ljcr.dmgordon.org/cover/show_cover.php?v=41&k=12&t=5 pending

Machine verification

Certificate check passed score 2207 2026-11-11 17:51 UTC

Valid (41,12,5) covering with 2207 blocks.

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