should know the irreducible characters of $Z_2^n$.

\[lem-z-3\] The irreducible characters of $Z_2^n$ are $\chi_{i_1,\ldots,i_n}$ for $i_j\in\{0,1\}$ and $1\le j\le n$, where $\displaystyle \chi_{i_1,\ldots,i_n}(v)=(-1)^{a_1i_1+\cdots+a_ni_n}$ for $v=(a_1,\ldots,a_n)^T\in Z_2^n$.

For a group $G$ and a subset $T$, denote by $\chi(T)=\sum_{t\in T}\chi(t)$ for a character $\chi$. By immediate calculation, we get the following result.

\[lem-z-4\] Let $\chi_{i_1,\ldots,i_n}$ be an irreducible character of $Z_2^n$ and $O_i$ an orbit. If $i_1+\cdots+i_n=k$, then $\displaystyle \chi_{i_1,\ldots,i_n}(O_i)=\sum_{j=0}^i{k\choose j}{n-k\choose i-j}(-1)^j$.

By definition, we have $\chi_{i_1,\ldots,i_n}(O_i)=\sum_{v\in O_i}\chi_{i_1,\ldots,i_n}(v)$. Denote by $\Lambda_j^i=\{v=(a_1,\ldots,a_n)^T\in O_i\mid a_1i_1+\cdots+a_ni_n=j\}$. Therefore, for each $v\in \Lambda_j^i$, we have $\chi_{i_1,\ldots,i_n}(v)=(-1)^j$ and thus $\chi_{i_1,\ldots,i_n}(O_i)=\sum_{j=0}^{r(i,k)}|\Lambda_j^i|(-1)^j$, where $r(i,k)=\min\{i,k\}$. It is not hard to verify that $|\Lambda_j^i|={k\choose j}{n-k\choose i-j}$. Thus, we have $$\displaystyle \chi_{i_1,\ldots,i_n}(O_i)=\sum_{j=0}^{r(i,k)}{k\choose j}{n-k\choose i-j}(-1)^j=\sum_{j=0}^i{k\choose j}{n-k\choose i-j}(-1)^j.$$

\[thm-z-1\] Let $Cay(Z_2^n,S)$ be an orbit Cayley graph over $Z_2^n$. If $S=\cup_{i\in I}O_i$ for some subset $I\subseteq\{1,\ldots,n\}$, then the eigenvalues of $Cay(Z_2^n,S)$ consist of $\lambda_k(S)$ with multiplicity ${n\choose k}$ for $0\le k\le n$, where $$\displaystyle \lambda_k(S)=\sum_{i\in I}\sum_{j=0}^{i}{k\choose j}{n-k\choose i-j}(-1)^j.$$

By Lemma \[lem-z-2\], the eigenvalues of $Cay(Z_2^n,S)$ are given by $$\chi_{i_1,\ldots,i_n}(S)=\sum_{i\in I}\chi_{i_1,\ldots,i_n}(O_i)$$ where $\chi_{i_1,\ldots,i_n}$ are irreducible characters of $Z_2^n$. Denote by $\Upsilon_k=\{\chi_{i_1,\ldots,i_n}\mid i_1+\cdots+i_n=k\}$ for $0\le k\le n$. By Lemma \[lem-z-4\], we have $$\chi_{i_1,\ldots,i_n}(S)=\sum_{i\in I}\sum_{j=0}^i{k\choose j}{n-k\choose i-j}(-1)^j$$ for any $\chi_{i_1,\ldots,i_n}\in \Upsilon_k$. It means that all characters in $\Upsilon_k$ 