Splittings of cyclic groups, tilings of Euclidean space, and perfect shift codes

Often in mathematics there is a deep interplay between algebra and geometry. So, splittings of groups correspond to lattice tilings of the Euclidean space by certain star bodies. This links two code concepts, which at first glance do not seem to be closely related: shift codes and codes in the Lee metric and error measures defined by similar spheres ("Stein sphere" and "Stein corner") around a codeword.