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

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Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

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≥m2+mn\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 KK of degree 4. We show that every line of a subplane of order 3 contained in KK lies in at most two such subplanes, whereas in PG(3,3)\mathrm{PG}(3,3), which like KK 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 AG(3,3)\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

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Version history

  1. v1 · 22 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23717956CurrentNov 12, 2026