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A $(k,k-t)$-SCID (set of Subspaces with Constant Intersection Dimension) is aset of $k$-dimensional vector spaces that have pairwise intersections ofdimension $k-t$. Let $\mathcal{C}=\{\pi_1,\ldots,\pi_n\}$ be a $(k,k-t)$-SCID.Define $S:=\langle \pi_1, \ldots, \pi_n \rangle$ and $I:=\langle \pi_i \cap\pi_j \mid 1 \leq i < j \leq n \rangle$. We establish several upper bounds for$\dim S + \dim I$ in different situations. We give a spectrum result for thecase $(n-1)(k-t)\leq k$ and for the case $n\leq\frac{q^{t(n-\eta)}-1}{q^t-1}$,giving examples of $(k,k-t)$-SCIDs reaching a large interval of values for$\dim S + \dim I$.

Properties of sets of Subspaces with Constant Intersection Dimension