  The Fuglede conjecture states that a set is spectral if and only if it tiles
by translation. The conjecture was disproved by T. Tao for dimensions 5 and
higher by giving a counterexample in $\mathbb{Z}_3^5$. We present a computer
program that determines that the Fuglede conjecture holds in $\mathbb{Z}_5^3$
by exhausting the search space. A. Iosevich, A. Mayeli and J. Pakianathan
showed that the Fuglede conjecture holds over prime fields when the dimension
does not exceed 2. The question for dimension 3 was previously addressed by
Aten et al. for $p=3$. In this paper we build upon the results of their work to
allow a computer to carry out the lengthy computations.
