2j+1}{4m-4k\choose 4t-2j-1}$ pairs of $(x,y)$ such that $v=x+y$ and $x,y\in S_0$, and there are $\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j+2}$ pairs of $(x,y)$ such that $v=x+y$ and $x,y\in S_3$. Therefore, by Theorem \[thm-z-3\] (xi), we have $$\begin{array}{lll}
|C(v,S_0\cup S_3)|&=&\displaystyle\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j-1}+\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j+2}\\
&=&\displaystyle\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k+1\choose 4t-2j+3}\\
&=&2^{4m}-(-1)^m2^{2m}.
\end{array}$$ It completes the proof.

Note that, from Lemma \[lem-z-7\] (f), $|S_0\cup S_3|=2^{4m+1}-(-1)^m2^{2m}-1$. Combining Lemmas \[lem-z-13\], \[lem-z-14\] and Corollary \[cor-z-3\], we get the following result.

\[thm-z-7\] The orbit Cayley graph $Cay(Z_2^{4m+2},S_0\cup S_3)$ is strongly regular with parameter $$\left(2^{4m+2},2^{4m+1}-(-1)^m2^{2m}-1,2^{4m}-(-1)^m2^{2m}-2,2^{4m}-(-1)^m2^{2m}\right)$$ for any positive integer $m$.

Since $\overline{Cay(Z_2^{4m+2},S_0\cup S_3)}=Cay(Z_2^{4m+2},S_1\cup S_2)$, we have the following result.

\[cor-z-6\] The orbit Cayley graph $Cay(Z_2^{4m+2},S_1\cup S_2)$ is strongly regular with parameter $$\left(2^{4m+2},2^{4m+1}+(-1)^m2^{2m},2^{4m}+(-1)^m2^{2m},2^{4m}+(-1)^m2^{2m}\right)$$ for any positive integer $m$.

Conclusion
==========

In this paper, we have constructed six families of infinite non-trivial strongly regular graphs from the general linear group $GL(n,F_2)$, see Theorems \[thm-z-5\], \[thm-z-6\], \[thm-z-7\] and Corollaries \[cor-z-4\], \[cor-z-5\] and \[cor-z-6\]. We collect them in Table \[tab-1\], and especially for $m=1$ and $2$, we present these graphs in Table \[tab-2\]. Note that Brouwer <http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html> has listed all strongly regular graphs on at most $1300$ vertices and thus the $12$ graphs in Table \[tab-2\] are contained in his collection. However, as we know, the six 