Almost perfect linear Lee codes of packing radius 2 only exist for small dimensions

TL;DR

This paper investigates the existence of almost perfect linear Lee codes with packing radius 2 in integer lattices, establishing that such codes only exist in specific small or special dimensions.

Contribution

It classifies the dimensions in which almost perfect linear Lee codes of packing radius 2 can exist, confirming their rarity and limited occurrence.

Findings

Almost perfect linear Lee codes of packing radius 2 only exist in specific dimensions.

The identified dimensions include small and special cases such as 1, 2, 11, 29, 47, 56, 67, 79, 104, 121, 134, 191.

Supports the conjecture that perfect Lee codes are extremely rare or nonexistent in higher dimensions.

Abstract

It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved this conjecture for linear Lee codes with $r=2$. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius $2$. We show that if such codes exist in $\mathbb{Z}^n$, then $n$ must be $1,2, 11, 29, 47, 56, 67, 79, 104, 121, 134$ or $191$.