Title:  Tiling n-space by unit cubes

A lattice tiling T of n-space by cubes is a tiling where the centers of cubes in T form a group under the vector addition. In 1907 Minkowski conjectured that in a lattice tiling of n-space by unit cubes there must be a pair of cubes that share a complete (n-1)-dimensional face. Minkowski's problem attracted a lot of attention as it is an interface of several mathematical disciplines. In fact, Minkowski's problem, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a$�+b�=c�$.

We discuss the conjecture, its history and variations, and then we describe some problems that Minkowski's conjecture, in turn, suggested. We will focus on tilings of n-space by clusters of cubes, namely by spheres in Lee metric, and show how these tilings correspond to the perfect error-correcting codes. The Golomb-Welch conjecture, the long-standing and the most famous conjecture in the area, will be discussed. An application of error-correcting codes to an optimization problem in computer architecture, as well as related algorithmic problems, will be presented.