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 also prove the conjecture in $\mathbb{Z}_2^4$, resolving all cases of four-dimensional vector spaces over prime fields. We give an example showing that our simple proof method does not extend to higher dimensions. However, we include a link to a computer program which the authors have used to verify that, nevertheless, the conjecture holds in $\mathbb{Z}_2^5$ and $\mathbb{Z}_2^6$ 查看全文>>