Equidistant Linear Codes in Projective Spaces

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 $q > 2$.