New lower bound for 2-identifying code in the square grid

TL;DR

This paper improves the lower bound on the density of 2-identifying codes in the square grid from approximately 0.162 to 0.171, narrowing the gap towards the known upper bound.

Contribution

It establishes a new lower bound of 6/35 for 2-identifying codes in the square grid, advancing the understanding of minimal code density.

Findings

Lower bound improved to 6/35 (~0.171)

Previous lower bound was 6/37 (~0.162)

Upper bound known as 5/29 (~0.172)

Abstract

An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ the intersection of $C$ and the ball of radius $r$ centered at $u$ is nonempty and unique. Previously, $r$-identifying codes have been studied in various grids. In particular, it has been shown that there exists a 2-identifying code in the square grid with density $5/29 \approx 0.172$ and that there are no 2-identifying codes with density smaller than $3/20 = 0.15$. Recently, the lower bound has been improved to $6/37 \approx 0.162$ by Martin and Stanton (2010). In this paper, we further improve the lower bound by showing that there are no 2-identifying codes in the square grid with density smaller than $6/35 \approx 0.171$.