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

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 linear combination of characters orthogonal with respect to $E$. We disprove Fuglede's conjecture in $\mathbb{Z}_{p}^{4}$ for all odd primes $p$, by using log-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 method does not extend to $\mathbb{Z}_{p}^{3}$. We also prove the conjecture in $\mathbb{Z}_{2}^{4}$, resolving all cases of four-dimensional vector spaces over prime fields. Our simple proof method does not extend to higher dimensions. The authors, however, have written a computer program to verify that 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 fails in $\mathbb{Z}_{2}^{10}$. Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$ is now resolved in all cases except when $d=3$ and $p\geq 11$, or when $p=2$ and $d=7,8,9$.