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Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$, $p$ a prime, says that a subset$E$ tiles $\mathbb{Z}_{p}^{d}$ by translation if and only if $E$ is spectral,meaning any complex-valued function $f$ on $E$ can be written as a linearcombination of characters orthogonal with respect to $E$. We disprove Fuglede'sconjecture in $\mathbb{Z}_{p}^{4}$ for all odd primes $p$, by usinglog-Hadamard matrices to exhibit spectral sets of size $2p$ which do not tile,extending the result of Aten et al. that the conjecture fails in$\mathbb{Z}_{p}^{4}$ for primes $p \equiv 3 \pmod 4$ and in$\mathbb{Z}_{p}^{5}$ for all odd primes $p$. We show, however, that our methoddoes not extend to $\mathbb{Z}_{p}^{3}$. We also prove the conjecture in$\mathbb{Z}_{2}^{4}$, resolving all cases of four-dimensional vector spacesover prime fields. Our simple proof method does not extend to higherdimensions. The authors, however, have written a computer program to verifythat the conjecture holds in $\mathbb{Z}_{2}^{5}$ and $\mathbb{Z}_{2}^{6}$.Finally, we modify Terry Tao's counterexample to show that the conjecture failsin $\mathbb{Z}_{2}^{10}$. Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$ is nowresolved in all cases except when $d=3$ and $p\geq 11$, or when $p=2$ and$d=7,8,9$.

Fuglede's conjecture fails in 4 dimensions over odd prime fields