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

TL;DR

This paper presents improved bounds for $r$-identifying codes in the hex grid, achieving sparser codes with density less than 5/(6r), surpassing previous constructions.

Contribution

The authors develop a new construction for $r$-identifying codes in the hex grid with lower density than prior methods.

Findings

Density less than 5/(6r) achieved

Outperforms previous density of approximately 8/(9r)

Provides explicit code construction methods

Abstract

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 $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than $5/(6r)$, which is sparser than the prior best construction which has density approximately $8/(9r)$.