On the smallest non-trivial tight sets in Hermitian polar spaces
Let \(\theta_r(q)=(q^{r+1}-1)/(q-1)\). Consider a polar space \({\mathcal P}\) of rank \(r\) over the finite field \(\mathbb{F}_q\) of order \(q\). An \textit{\(x\)-tight set} of \({\mathcal P}\) is a subset \(T\) of the point set of \({\mathcal P}\) such that \(q^{r-1}+x\theta_{r-2}(q)\) points of \(T\) belong to \(P^\perp\) when \(P\in T\) and such that \(x\theta_{r-2}(q)\) points of \(T\) belong to \(P^\perp\) when \(P\not \in T\). The integer \(x\) is called the \textit{parameter} of the tight set \(T\). The classical example of an \(x\)-tight set \(T\) is a union of \(x\) pairwise disjoint generators of the polar space \({\mathcal P}\), when this union of \(x\) pairwise disjoint generators exists. The authors prove in this article the following two characterization results: every tight set in the Hermitian variety H\((4,q^2)\), with parameter \(x<q+1\), is a union of \(x\) lines of H\((4,q^2)\), and every tight set of the Hermitian variety H\((6,q^2)\), with parameter \(x\leq q+1-\sqrt{2q}\), is a union of \(x\) pairwise disjoint generators of H\((6,q^2)\). The interesting fact about the characterization result on the tight sets of the Hermitian variety H\((4,q^2)\) is that the result is sharp. There exist \((q+1)\)-tight sets on H\((4,q^2)\), not equal to the union of \(q+1\) lines. This is also explicitly mentioned in the article in Theorem 4.1: Let \(q\) be odd and \(n\geq 2\), then the set of points of a polar space \(Q(2n,q)\), \(q\) odd, embedded in the Hermitian variety H\((2n,q^2)\), is a \((q+1)\)-tight set of H\((2n,q^2)\).
scientific article; zbMATH DE number 6703754