A linear programming approach to Fuglede's conjecture in ${\mathbb{Z}}_p^3$

We present an approach to Fuglede’s conjecture in ${\mathbb{Z}}_p^3$ using linear programming bounds, obtaining the following partial result: if $A\subseteq {\mathbb{Z}}_p^3$ with $p^2-p\sqrt{p}+\sqrt{p}<\mathrm{\mid }A\mathrm{\mid }<p^2$, then $A$ is not spectral.