On regular systems of finite classical polar spaces

TL;DR

This paper investigates regular systems in finite classical polar spaces, establishing non-existence results, describing methods to generate new systems, and presenting three construction techniques for regular systems related to points.

Contribution

It introduces new non-existence results, a procedure to derive new regular systems from existing ones, and discusses three construction methods for regular systems in polar spaces.

Findings

Proved non-existence of certain 1-regular systems with low rank.

Developed a procedure to generate new m'-regular systems from existing m-regular systems.

Presented three methods for constructing regular systems related to points in polar spaces.

Abstract

Let $\cal P$ be a finite classical polar space of rank $d$. An $m$-regular system with respect to $(k - 1)$-dimensional projective spaces of $\cal P$, $1 \le k \le d - 1$, is a set $\cal R$ of generators of $\cal P$ with the property that every $(k - 1)$-dimensional projective space of $\cal P$ lies on exactly $m$ generators of $\cal R$. Regular systems of polar spaces are investigated. Some non-existence results about certain 1-regular systems of polar spaces with low rank are proved and a procedure to obtain $m'$-regular systems from a given $m$-regular system is described. Finally, three different construction methods of regular systems w.r.t. points of various polar spaces are discussed.