Perfect codes in the lp metric

TL;DR

This paper explores the existence and bounds of perfect codes in integer lattices under the p metric, providing new non-existence results and characterizations for specific dimensions and metrics.

Contribution

It derives upper bounds for the packing radius of linear perfect codes in p metric and determines all such codes for certain small dimensions and p values.

Findings

Upper bounds for packing radius in p metric

Complete classification of linear perfect codes for p with n=2,3 when p=2

Non-existence results for codes in n, impacting finite alphabet code constructions

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.