Perfect codes in the lplp metric

We investigate perfect codes in ZnZn in the ℓpℓp metric. Upper bounds for the packing radius rr of a linear perfect code, in terms of the metric parameter pp and the dimension nn are derived. For p=2p=2 and n=2,3n=2,3, we determine all radii for which there exist linear perfect codes. The non-existence results for codes in ZnZn presented here imply non-existence results for codes over finite alphabets ZqZq, when the alphabet size is large enough, and have implications on some recent constructions of spherical codes.