Lower bounds for identifying codes in some infinite grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $\mathrm{\mid }C\mathrm{\mid }\mathrm{/}\mathrm{\mid }V(G)\mathrm{\mid }$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.