Equidistant Codes in the Grassmannian

Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Pl"{u}cker embedding, for 1-intersecting codes of $k$-dimensional subspaces over ${\text{F}}_q^n$, $n\ge \left(\genfrac{}{}{0px}{}{k+1}{2}\right)$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n\times \left(\genfrac{}{}{0px}{}{n}{2}\right)$ over ${\text{F}}_q$, rank $n-1$, and rank distance $n-1$.