  Linear codes in the projective space $\mathbb{P}_q(n)$, the set of all
subspaces of the vector space $\mathbb{F}_q^n$, were first considered by Braun,
Etzion and Vardy. The Grassmannian $\mathbb{G}_q(n,k)$ is the collection of all
subspaces of dimension $k$ in $\mathbb{P}_q(n)$. We study equidistant linear
codes in $\mathbb{P}_q(n)$ in this paper and establish that the normalized
minimum distance of a linear code is maximum if and only if it is equidistant.
We prove that the upper bound on the size of such class of linear codes is
$2^n$ when $q=2$ as conjectured by Braun et al. Moreover, the codes attaining
this bound are shown to have structures akin to combinatorial objects, viz.
\emph{Fano plane} and \emph{sunflower}. We also prove the existence of
equidistant linear codes in $\mathbb{P}_q(n)$ for any prime power $q$ using
\emph{Steiner triple system}. Thus we establish that the problem of finding
equidistant linear codes of maximum size in $\mathbb{P}_q(n)$ with constant
distance $2d$ is equivalent to the problem of finding the largest
$d$-intersecting family of subspaces in $\mathbb{G}_q(n, 2d)$ for all $1 \le d
\le \lfloor \frac{n}{2}\rfloor$. Our discovery proves that there exist
equidistant linear codes of size more than $2^n$ for every prime power $q > 2$.
