C(v,S_0\cup S_1)$ such that $x\in S_0$ and $y\in S_1$. By symmetry of $x$ and $y$, there are also $\sum_{t=0}^m\sum_{j=0}^{2k}{4k+1\choose 2j}{4m-4k+1\choose 4t-2j}-1$ pairs of $(x,y)\in C(v,S_0\cup S_1)$ such that $y\in S_0$ and $x\in S_1$. Therefore, from Theorem \[thm-z-3\] (v), we also have $$|C(v,S_0\cup S_1)|=2\sum_{t=0}^m\sum_{j=0}^{2k}{4k+1\choose 2j}{4m-4k+1\choose 4t-2j}-2=2^{4m}+(-1)^m2^{2m}-2.$$

It completes the proof.

\[lem-z-12\] Let $v$ be an element of $Z_2^{4m+2}$. If $v\in \overline{S_0\cup S_1}=S_2\cup S_3$, then $$|C(v,S_0\cup S_1)|=2^{4m}+(-1)^m2^{2m}.$$

As similar as the proof of Lemma \[lem-z-10\], if $v\in S_2$, there are $\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j-1}+\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j}$ pairs of $(x,y)\in C(v,S_0\cup S_1)$. By Theorem \[thm-z-3\] (vi), we have $$\begin{array}{lll}
|C(v,S_0\cup S_1)|&=&\displaystyle\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j-1}+\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k\choose 4t-2j}\\
&=&\displaystyle\sum_{t=0}^m\sum_{j=0}^{2k}{4k+2\choose 2j+1}{4m-4k+1\choose 4t-2j}\\
&=&2^{4m}+(-1)^m2^{2m}.
\end{array}$$

If $v\in S_3$, there are $2\sum_{t=0}^m\sum_{j=0}^{2k+1}{4k+3\choose 2j+1}{4m-4k-1\choose 4t-2j-1}$ pairs of $(x,y)\in C(v,S_0\cup S_1)$. By Theorem \[thm-z-3\] (vii), we also have $$|C(v,S_0\cup S_1)|=2\sum_{t=0}^m\sum_{j=0}^{2k+1}{4k+3\choose 2j+1}{4m-4k-1\choose 4t-2j-1}=2^{4m}+(-1)^m2^{2m}.$$

It completes the proof.

Note that, from Lemma \[lem-z-7\] (g), $|S_0\cup S_1|=\sum_{i=1}^m{4m+2\choose 4i}+\sum_{i=0}^m{4m+2\choose 4i+1}=\sum_{i=0}^m\left({4m+2\choose 4i}+{4m+2\choose 4i+1}\right)-1=\sum_{i=0}^m{4m+3\choose 4i+1}=2^{4m+1}+(-1)^m2^{2m}-1$. Combining Lemmas \[lem-z-11\], \[lem-z-12\] and Theorem \[cor-z-3\], we get the following result.

\[thm-z-6\] The orbit Cayley graph $Cay(Z_2^{4m+2},S_0\cup S_1)$ 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 $\overline{Cay(Z_2^{4m+2},S_0\cup S_1)}=Cay(Z_2^{4m+2},S_2\cup S_3)$, we have the following result.

\[cor-z-5\] The orbit Cayley graph $Cay(Z_2^{4m+2},S_2\cup S_3)$ 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 