math.CO — Combinatorics

Subplanes of order 3 and maximal arcs in projective planes of order 12

Contributed by Tejasvi Singh Tomar

Whether a projective plane of order 12 exists is open. A subplane of order 3 of such a plane is a case of equality in Bruck's bound $n\ge m^2+m$; the 27 points on no line of the subplane then form a maximal arc of degree 3, and together with the subplane a maximal arc $K$ of degree 4. We show that every line of a subplane of order 3 contained in $K$ lies in at most two such subplanes, whereas in $\mathrm{PG}(3,3)$, which like $K$ has 40 points, a line lies in four planes; the proof uses Egan and Wanless's classification of triples of mutually orthogonal Latin squares of order 9. We also show that $\mathrm{AG}(3,3)$ is not the design of any maximal arc of degree 3 in a plane of order 12, and that a permutation of the points that maps a subplane of order 3 onto itself and each line meeting it onto a line cannot have prime order other than 3. The proofs are computer-assisted, and their SAT refutations are certified.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23717956