  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 $\F_q^n$,
$n \geq \binom{k+1}{2}$, 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 $n \times \binom{n}{2}$
over $\F_q$, rank $n-1$, and rank distance $n-1$.
