Experimental Results of the Search for Unitals in Projective Planes of Order 25

TL;DR

This paper reports on a computational search for unitals within known projective planes of order 25, revealing the existence of multiple nonisomorphic unitals and some isomorphic to those in other planes, with many planes lacking found unitals.

Contribution

The paper provides the first extensive computational enumeration of unitals in projective planes of order 25, identifying their isomorphism classes and occurrence patterns.

Findings

Multiple nonisomorphic unitals found per plane.

Some unitals are isomorphic across different planes.

Many planes have no detected unitals.

Abstract

In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one unitals have been found for most of the planes. The found unitals for a given plane are nonisomorphic each other. There are a few unitals isomorphic to a unital of another plane. A t - (v; k; {\lambda}) design D is a set X of points together with a family B of k-subsets of X called blocks with the property that every t points are contained in exactly {\lambda} blocks. The design with t = 2 is called a block-design. The block-design is symmetric if the role of the points and blocks can be changed and the resulting confguration is still a block-design. A projective plane of order n is a symmetric 2-design with v = n2 + n + 1, k = n + 1, {\lambda} = 1. The blocks of such a design are called lines. A unital in a projective plane of order n = q2 is a set U of q3 + 1 points that meet every line in one or q + 1 points. In the case projective planes of order n = 25 we have: q = 5, the projective plane is 2 - (651; 26; 1) design, the unital is a subset of q3 + 1 = 53+ 1 = 126 points and every line meets 1 or 6 points from the subset