Properties of sets of subspaces with constant intersection dimension
Lisa Hernandez Lucas · Advances in Mathematics of Communications · 2020
A \begin{document}$ (k,k-t) $\end{document} -SCID (set of Subspaces with Constant Intersection Dimension) is a set of \begin{document}$ k $\end{document} -dimensional vector spaces that have pairwise intersections of dimension \begin{document}$ k-t $\end{document} . Let \begin{document}$ \mathcal{C} = \{\pi_1,\ldots,\pi_n\} $\end{document} be a \begin{document}$ (k,k-t) $\end{document} -SCID. Define \begin{document}$ S: = \langle \pi_1, \ldots, \pi_n \rangle $\end{document} and \begin{document}$ I: = \langle \pi_i \cap \pi_j \mid 1 \leq i < j \leq n \rangle $\end{document} . We establish several upper bounds for \begin{document}$ \dim S + \dim I $\end{document} in different situations. We give a spectrum result under certain conditions for \begin{document}$ n $\end{document} , giving examples of \begin{document}$ (k,k-t) $\end{document} -SCIDs reaching a large interval of values for \begin{document}$ \dim S + \dim I $\end{document} .