Fuglede's conjecture holds in $\mathbb{F}_2^7$

Abstract

We prove that a nonempty subset of the binary vector space $\mathbb{F}_2^7$ is spectral if and only if it tiles $\mathbb{F}_2^7$ by translations. This is the smallest binary dimension left open by Ferguson and Sothanaphan. For the implication from tiling to spectrality we show that every eight-point subset of $\mathbb{F}_2^d$ admits $2^{d-3}$ frequencies whose pairwise differences avoid the zeros of its Fourier transform; consequently every tile of $\mathbb{F}_2^d$ with at least $2^{d-3}$ elements is spectral. For the converse implication, we use the Smith normal form of the Walsh matrix to exclude every spectral subset of $\mathbb{F}_2^7$, and of $\mathbb{F}_2^8$, whose size is not a power of two. The sizes $16$ and $32$ are reduced, by a projection argument, to the tiling of point sets attached to binary log-Hadamard matrices of small rank. Using the published classifications of Hadamard matrices of orders $16$ and $32$, we find that exactly sixteen order-$32$ classes up to transposition have dephased binary rank at most seven, and we give explicit tilings for both orientations of each.