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We present an approach to Fuglede's conjecture in $\mathbb{Z}_p^3$ usinglinear programming bounds, obtaining the following partial result: if$A\subseteq\mathbb{Z}_p^3$ with $p^2-p\sqrt{p}+\sqrt{p}<|A|<p^2$, then $A$ isnot spectral.

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