Perfect truncated-metric codes in lattice graphs and superlattice graphs
Italo Jose Dejter · Journal of Combinatorial Mathematics and Combinatorial Computing · 2025
Perfect codes in the \(n\)-dimensional grid \(\Lambda_n\) of the lattice \(\mathbb{Z}^n\) (\(0 ) 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 superlattice 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 \(n\).