Let $K$ be an algebraic number field and ${𝒪}_K$ the ring of integers of $K$. In this paper, we prove an analogue of Voronoï’s theorem for ${𝒪}_K$-lattices and the finiteness of the number of similar isometry classes of perfect ${𝒪}_K$-lattices.

We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant ${\gamma }_{n,k}$ and the Bergé–Martinet constant ${\gamma }_{n,k}^{\prime }$. Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of ${\gamma }_5^{\prime },{\gamma }_7^{\prime }$, ${\gamma }_{4,2}$ and ${\gamma }_n$ ($n\leqq 8$).

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group ${\mathbf{SL}}_2\left({\mathbf{F}}_{41}\right)\otimes {\tilde{S}}_3\subset {\mathbf{GL}}_{80}\left(\mathbf{Z}\right)$ is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the $\Theta$-series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner product...