Perfect codes in the lp metric

Abstract

We investigate perfect codes in ${\mathbb{Z}}^n$ under the ${\ell }_p$ metric. Upper bounds for the packing radius $r$ of a linear perfect code, in terms of the metric parameter $p$ and the dimension $n$ are derived. For $p=2$ and $n=2,3$, we determine all radii for which there are linear perfect codes. The non-existence results for codes in ${\mathbb{Z}}^n$ presented here imply non-existence results for codes over finite alphabets ${\mathbb{Z}}_q$, when the alphabet size is large enough, and has implications on some recent constructions of spherical codes.

21 pages, 9 figures, minor corrections, accepted for publication European Journal of Combinatorics

References in corpus (1)

• A New Approach Towards the Golomb-Welch Conjecture