Packings of by certain error spheres

Golomb in 1969 defined error metrics for codes and their corresponding "error spheres." Among the error spheres are the cross and semicross. The cross is defined as follows. Let and be positive integers. The -cross in Euclidean -space consists of unit cubes: a central cube together with arms of length . The -semicross in consists of unit cubes: a comer cube together with arms of length attached at of its nonopposite faces. For instance, the -cross has five squares arranged in a cross and the -semicross is shaped like the letter . Much has been done on determining when translates of a cross or semicross tile (or tesselate) . If translates do not tile, we may ask how densely they can pack space without overlapping. We answer this question for the -cross in all dimensions and for large. We also show that packings by the cross that are extremely regular (lattice packings) do just about as well as arbitrary packings by the cross. However, for the semicross, even in , when the arm length is large, lattice packings are much less dense than arbitrary packings. The methods are primarily algebraic, involving Abelian groups.