Abstract
Start with the $1$-dimensional lattice $\Lambda_1$ of even integer points: at the $n$-th step stack layers of a suitable $(n-1)$-dimensional lattice $\Lambda_{n-1}$ as densely as possible, keeping the same minimal norm; the result is a laminated lattice $\Lambda_n$. In this paper the density of $\Lambda_n$ is determined for $n\le 48$, all $\Lambda_n$ are found for $n\le 25$, and at least one $\Lambda_n$ is found for $26 \le n\le 48$. The unique $\Lambda_{24}$ is the Leech lattice.
For completeness all the best sphere-packings known in dimensions up to $48$ are described in the paper. A number of these are new.
