On perfect q-ary codes in the maximum metric

Perfect codes in a given metric are the ones which attain the sphere packing bound. In the Hamming metric perfect linear codes over finite fields are known only to exist for a restricted number of parameters. In the Lee metric, perfect linear codes in Z
ⁿ
are classified for n = 2 and the long standing Golomb-Welch conjecture that there are no perfect codes for n ≥ 3, except for packing radius 1, still remains open. We approach here perfect codes over finite rings in the maximum metric (also known as L
_∞
or Chebyshev metric). This metric has been recently considered in coding schemes for flash memory. A perfect linear code in Z
ⁿ
_q
in the maximum metric corresponds to an integer lattice cube tiling. We present a complete classification of two-dimensional perfect codes, constructions of perfect codes from codes of smaller dimensions and generator matrices for n-dimensional perfect linear q-ary codes in the maximum metric. From these results a description or all perfect codes in the Euclidean metric for n = 2, 3 is derived.