On Gabor orthonormal bases over finite prime fields

We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existence of a Gabor window $g$ with full support where neither $|g|$ nor $|\hat g|$ is an indicator function and $|B|<<p^d$. We conclude the paper with an example and open questions.