TejasviTomar
October 2026 · Preprint

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

Tejasvi Singh Tomar

In plain words

In a finite projective plane, two points share exactly one line, and two lines exactly one point. A plane's order is one less than the number of points on a line. Planes exist for every order that is a power of a prime, and none is known for any other order. A large computer search ruled out order 10 in 1989, leaving 12 as the smallest order still open. The paper works out what a plane of order 12 containing one of order 3 would look like, and rules out several forms the larger plane could take.

Abstract

The existence of a projective plane of order 12 is open. A subplane of order 3 of such a plane attains Bruck's bound n ≥ m² + m with equality. The counts in this equality case show that the 27 points on no line of the subplane form a maximal arc of degree 3, and that together with the subplane they form a maximal arc of degree 4. Its secants carry a Steiner system S(2,4,40), and Thas's construction yields partial geometries pg(8,9,6) and pg(9,8,6), whose existence is listed as open. If the 27 points form the affine space AG(3,3) with the tangent classes as directions, the plane extends the points and lines of PG(3,3), and Hall and Roth, and Prince, showed that no such plane exists. We show that in this case already three lines of the subplane lead to a contradiction. More generally, AG(3,3) is not the design of a maximal arc of degree 3 in any plane of order 12. The incidence structure formed by all points and the lines that meet the subplane has no automorphism of prime order other than 3 that preserves the subplane, and one type of order 3 is excluded as well. For maximal arcs of degree 3 or 4, with no subplane assumed, the analogous structure has no automorphism of order 13. With the classification of Kirkman triple systems of order 27 with an automorphism of order 13 by Janko and van Trung, whose count we correct from 661 to 657, we also exclude the case where the 27 points carry such an automorphism compatible with the labelling of their parallel classes. Finally, suppose that every line ℓ of the subplane splits the 27 points into three sets of 9, two points lying in the same set exactly when their joining line meets ℓ in a point of the subplane. Then the 27 points form a Hall triple system, hence AG(3,3), and the plane does not exist. The computer-assisted steps, including a proof of Prince's theorem that PG(3,3) has no covering set of spreads, are checked in Lean 4: the SAT refutations through LRAT proofs, and the remaining exhaustive searches by search procedures whose completeness is proved in Lean.

Thirteen dark dots evenly spaced on a circle, with no labels, joined by thin teal lines that form thirteen overlapping quadrilaterals in a rotating pattern. One quadrilateral, through four of the dots, is drawn in bold magenta with a pale pink fill, and its four dots are magenta.
Illustration: the projective plane of order 3 (the subplane of the title), drawn with its 13 points on a circle. Each of the 13 lines is drawn as the quadrilateral joining its four points; one of the lines is shown in magenta. Any two points lie on exactly one line, and any two lines meet in exactly one point. Turning the picture by one step, so that each point moves to the next, maps lines to lines. In a plane of order 12 containing this one, 27 points would lie on none of the subplane’s lines. Janko and van Trung showed in 1980 that no symmetry of the larger plane can turn the subplane by one step; the paper proves this even for maps that need to preserve only the lines that meet the subplane.