cyclic groups. The complete characterization of strongly regular circulant graphs are independently given by Bridges and Mena [@Bridges], Ma [@Ma], and partially by Marušič [@Marusic]. For other abelian groups, Leifman and Muzychuk [@Leifman] classified strongly regular Cayley graphs on $Z_{p^n}\times Z_{p^n}$ for a prime $p$. Recently, the complete characterization of *minimal Cayley graphs* (that is Cayley graphs $Cay(G,S)$ such that $S$ is a minimal generating set of $G$) over abelian groups was given by [@Stefko].

In this paper, we focus on the Cayley graphs over the elementary abelian $2$-group $Z_2^n$. It is clear that the automorphism group of $Z_2^n$ is isomorphic to the general linear group $GL(n,F_2)$. Let $W$ be the collection of permutation matrices of order $n$. It is clear that $W\le GL(n,F_2)$. Note that $Z_2^n$ can be viewed as the vector space of dimension $n$ over the field $F_2$. Therefore, each element $v$ in $Z_2^n$ is a vector $v=(a_1,\ldots,a_n)^T$ such that $a_i\in F_2$. Let $|v|$ be the number of $1$’s in $v$, that is, $|v|=\sum_{i=1}^na_i$. Denote by $O_i=\{v\in Z_2^n\mid |v|=i\}$ for $0\le i\le n$. It is clear that $Pv\in O_i$ for any $P\in W$ and $v\in O_i$. Moreover, for any two vectors $u,v\in O_i$, there exists $P\in W$ 