# Optimal local identifying and local locating-dominating codes

## TL;DR

This paper introduces new classes of covering codes called local r-identifying and local r-locating-dominating codes, analyzing their sizes and densities in various graph structures.

## Contribution

It defines and studies the properties of local identifying and locating-dominating codes, providing bounds, constructions, and density results in different grids.

## Key findings

- Asymptotically tight bounds for optimal local 1-identifying codes in hypercubes.
- Linear code construction achieves the upper bound for these codes.
- Most of the constructed codes have optimal densities in various grid graphs.

## Abstract

We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities.