Multilattice graphs and perfect domination

TL;DR

This paper explores perfect codes in multilattice graphs constructed from ternary cubes, extending known lattice codes and conjecturing their existence in higher dimensions with larger radii.

Contribution

It introduces multilattice graphs based on ternary cubes and demonstrates the existence of infinite isolated perfect truncated-metric codes for dimension 2, proposing their potential existence in higher dimensions.

Findings

Existence of infinite perfect codes in 2D multilattice graphs.

Construction method for multilattice graphs from ternary cubes.

Conjecture on the existence of such codes for higher dimensions.

Abstract

Perfect codes in the $n$-dimensio\-nal grid $\Lambda_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and $v=(v_1, \ldots,v_n)$ as the graph distance $h(u,v)$ in $\Lambda_n$, if $|u_i-v_i|\le 1$, for all $i\in\{1, \ldots,n\}$, and as $n+1$, otherwise. Such codes are extended to multilattice graphs $\Gamma_n$ obtained by glueing ternary $n$-cubes along their codimension 1 ternary subcubes in such a way that each binary $n$-subcube is contained in a unique maximal lattice of $\Gamma_n$. The existence of an infinite number of isolated perfect truncated-metric codes of radius 2 in $\Gamma_n$ for $n=2$ is ascertained, leading to conjecture such existence for $n>2$ with radius $n$.