Tilings by $(0.5,n)$-Crosses and Perfect Codes

TL;DR

This paper investigates tilings of Euclidean space by scaled crosses with arms of length 0.5, establishing existence conditions linked to perfect codes and connecting to binary and ternary Hamming codes.

Contribution

It proves that integer tilings by these scaled crosses exist only in dimensions of the form 2^t-1 or 3^t-1, revealing a connection to perfect codes.

Findings

Existence of tilings only in dimensions 2^t-1 or 3^t-1.

Strong link between these tilings and binary/ternary perfect codes.

Provides new insights into geometric and coding theory relationships.

Abstract

The existence question for tiling of the $n$-dimensional Euclidian space by crosses is well known. A few existence and nonexistence results are known in the literature. Of special interest are tilings of the Euclidian space by crosses with arms of length one, known also as Lee spheres with radius one. Such a tiling forms a perfect code. In this paper crosses with arms of length half are considered. These crosses are scaled by two to form a discrete shape. We prove that an integer tiling for such a shape exists if and only if $n=2^t-1$ or $n=3^t-1$, $t>0$. A strong connection of these tilings to binary and ternary perfect codes in the Hamming scheme is shown.