Perfect truncated-metric codes and a union of ternary squares expanding the square lattice

Perfect truncated-metric codes (PTMCs) in the $n$-dimensio-nal grid $Lambda_n$ of $mathbb{Z}^n$ ($0<ninmathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance $rho(u,v)$ in $mathbb{Z}^n$ given between vertices $u=(u_1,cdots,u_n)inmathbb{Z}^n$ and $v=(v_1, ldots,v_n)inmathbb{Z}^n$ as the Hamming distance $h(u,v)$ in $mathbb{Z}^n$ (or graph distance $h(u,v)$ in $Lambda_n$) if $|u_i-v_i|le 1$, for all $iin{1, ldots,n}$, and as $n+1$, otherwise. While this $rho$ is related to the $ell_p$ metrics, PTMCs associated with lattice tilings of $mathbb{Z}^n$ were recently worked upon as rainbow perfect dominating sets. Now, PTMCs are extended to ternary compounds $Gamma_n$ obtained by glueing, or locking, ternary $n$-cubes along their codimension 1 ternary subcubes. Such compounds may be taken as alternate reality graphs, since they offer a third option in each coordinate direction at each vertex. We ascertain the existence of an infinite number of isolated PTMC of radius 2 in $Gamma_n$ for $n=2$ and conjecture such existence for $n>2$ with radius $n$. We ask whether there exists a suitable notion replacing that of quotient toroidal grids of $Lambda_n$ for the case of $Gamma_n$.