A non-lattice periodic point set beating the optimal lattice packing-covering constant in dimension five

The packing-covering constant of a point set $X\subseteq {\mathbb{R}}^d$ is $\gamma (X)=\mu (X)\mathrm{/}\rho (X)$, the covering radius divided by the packing radius. Among lattices, its minimum ${\gamma }_d$ is known for $d\le 5$, attained by ${\mathsf{A}}_2^{\ast }$, ${\mathsf{A}}_3^{\ast }$, and Horváth’s lattices ${\mathsf{H}\mathsf{o}}_4$, ${\mathsf{H}\mathsf{o}}_5$; Böröczky proved that ${\gamma }_3$ is optimal without the lattice restriction, but for $d=4,5$ the non-lattice problem was open. We exhibit a $2$-periodic non-lattice point set of ${\mathbb{R}}^5$ with $\gamma =9\mathrm{/}\sqrt{40}=1.423024\dots <{\gamma }_5=\sqrt{3\mathrm{/}2+\sqrt{13}\mathrm{/}6}=1.4494568\dots$, so that in dimension five the packing-covering problem is not solved by lattices.