Improved Bounds for $r$-Identifying Codes of the Hex Grid

For any positive integer $r$, 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 pairwise distinct. For 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 find a code of density less than $5\mathrm{/}(6r)$, which is sparser than the prior best construction which has density approximately $8\mathrm{/}(9r)$.