Perfect truncated-metric codes in lattice graphs and superlattice graphs

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