we also have PGDS with parameters $(v=3^{n}, k=3^{n-1}, \alpha=3^{2n-3}-3^{n-2}, \beta=3^{n-1}+3^{2n-3}-3^{n-2})$. Here we also note that not all decimation will lead such a partition of the finite field $ {{\mathbb F}}_{3^n}$. For instance, let $D_i=\{x: F(x)=Tr_n(x^d)=i\}$ for $d=3^{2}-3+1$. Then computational results imply that none of the $D_i$’s is a partial geometric difference set in ${{\mathbb F}}_{3^5}$. In general, it is a challenging task to characterize all functions which can be used to obtain a partition of a group into partial geometric difference sets.

If there is such a partition we can define a set of complex vectors $$z_{a}=(\chi_a(D_0),\chi_a(D_1),\chi_a(D_2))$$ for any $a\in \mathbb{F}_{3^n}$. Let $$e=(1,\zeta_3, \zeta_3^2).$$ Then norm of the complex inner product of any vector $z_{a}$ and $e$ is either $0$ or $C$ for some integer $C$.

Let $D_0$, $D_1$ and $D_2$ be a partition of $\mathbb{F}_{3^n}$ and $\lambda=3^{\frac{n+s-1}{2}}$ be an integer. Suppose for $i=0,1,2$ each $D_i$ is a partial geometric difference set such that $\chi_a(D_i) \in  \{0, \pm \lambda,\pm \lambda \zeta_3,\pm \lambda \zeta_3^2\}$ for each nonprincipal character $\chi_a$. If one of the cases holds

-   $|D_0|=|D_1|=|D_2|$,

-   $|D_i|=|D_j|=3^{n-1}-3^{\frac{n+s-2}{2}}$ and $|D_k|=3^{n-1}+2.3^{\frac{n+s-2}{2}}$,

-   $|D_i|=|D_j|=3^{n-1}+3^{\frac{n+s-2}{2}}$ and $|D_k|=3^{n-1}-2.3^{\frac{n+s-2}{2}}$.

and $|<z_a,e>|^2$ is either $0$ or $3\lambda^2$ then $$f(x)=\begin{cases} 
    0, & x \in D_0\\
    1, & 