The Non-Existence of Finite Projective Planes of Order 10

Abstract

This note reports the result of a computer search which shows that no 19-point con guration can be extended to a complete nite projective plane of order 10. Previous work has shown that such a plane, if it exists, must contain 24,675 19-point con gurations. Together, these results imply the non-existence of a projective plane of order 10. This note presents a brief summary of the previous work and gives some details of the methodology which are not covered by earlier publications. It also argues that, even when the possibility of undetected software or hardware errors is taken into account, the probability is very small that an undiscovered plane of order 10 is missed by all the computer searches.

XSAT and NAE-SAT of linear CNF classes

“…A k-block formula directly corresponds to a finite projective plane of order k −1 [1]. Unfortunately it is a hard open question to decide whether a projective plane exists for a given positive integer k [15,16]. However, it is a well known fact in combinatorics that for prime power orders the corresponding projective planes can easily be computed [15].…”

Designs, Groups and Computing

“…The first unsettled case is n = 10. It is a celebrated computational result that there is no projective plane of order 10 [22,28], so there do not exist nine MOLS of order 10, and so a uniform (10 × 10)/9 semi-Latin square does not exist. On the other hand, SLS(PSL 2 (9)) and inflations of this square yield uniform (10 × 10)/(9µ) semi-Latin squares for µ = 4, 8, 12, 16, .…”

Using Volunteer Computing to Study Some Features of Diagonal Latin Squares

“…For example, the fact, that there is no nite projective plane of order 10, was proven on a computing cluster [15]. The hypothesis about the minimal number of clues in Sudoku was rst proven on a computing cluster too [19].…”