Abstract:

We investigate the class of completely regular codes in graphs with a distance partition $C_0, \ldots, C_{\rho}$, where each set $C_i$, for $0 \leq i \leq s - 1$, is an independent set. This work focuses on the existence problem for such codes in the $n$-dimensional infinite grid. We demonstrate that several parameter families of such codes necessarily arise from binary or ternary Hamming graphs or  do not exist. Furthermore, employing binary linear programming techniques, we explore completely regular codes in infinite grids of dimensions $3$ and $4$ for the cases $s = 1$ and $s = 2$.