Existence and non-existence of \(m\)-systems of polar spaces
A partial \(m\)-system, \(m \geq 0\), of a finite polar space \({\mathcal P}\) of rank \(r \geq \max(2,m+1)\) is a set \(\{ \pi_1,\ldots,\pi_k \}\) of \(k \not=0\) totally singular \(m\)-spaces such that no generator containing \(\pi_i\) has a point in common with \((\pi_1 \cup \ldots \cup \pi_k) \setminus \pi_i\), \(i=1,\ldots,k\). If the upper bound for \(k\) derived by \textit{E. E. Shult} and \textit{J. A. Thas} [J. Comb. Theory, Ser. A 68, No. 1, 184-204 (1994; Zbl 0824.51004)] is obtained then the partial \(m\)-system is called an \(m\)-system. In the paper the authors obtain non-existence results for \(m\)-systems by examining the properties of certain strongly regular graphs associated with the point-sets of the \(m\)-systems. Using the computer, they also classify all \(m\)-systems of \(W_{2n+1}(2)\), \(Q(2n+2,2)\), \(Q^-(2n+1,2)\) and \(Q^+(2n+1,2)\), \(n \in \{ 1,2,3,4 \}\).
scientific article; zbMATH DE number 1567133
scientific article