Year: 2026
Type: preprint
Abstract: We prove that a nonempty subset of the binary vector space F27 is spectral if and only if it tiles F27 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 F2d admits 2d−3 frequencies whose pairwise differences avoid the zeros of its Fourier transform; consequently every tile of F2d with at least 2d−3 elements is spectral. For the converse implication, we use the Smith normal form of the Walsh matrix to exclude every spectral subset of F27, and of F28, 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. The supplementary archive contains the computational certificates, selected representatives, verification programs, and reproduction instructions.
Source: arXiv (Cornell University)
Author: Kadowaki, Atsushi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220916819