Quasi perfect codes in the cartesian product of some graphs

An important question in the study of quasi-perfect codes is whether such codes can be constructed for all possible lengths $n$. In this paper, we address this question for specific values of $n$. First, we investigate the existence of quasi-perfect codes in the Cartesian product of a graph $G$ and a path (or cycle), assuming that $G$ admits a perfect code. Second, we explore quasi-perfect codes in the Cartesian products of two or three cycles, $C_m\mathrm{\square }{C}_n$ and $C_m\mathrm{\square }{C}_n\mathrm{\square }{C}_l$, as well as in the Cartesian products of two or three paths, $P_m\mathrm{\square }{P}_n$ and $P_m\mathrm{\square }{P}_n\mathrm{\square }{P}_l$.