# Finding codes on infinite grids automatically

## TL;DR

This paper introduces a novel automata-based approach to improve bounds on the minimum density of identifying codes in infinite grids, achieving a new record for the hexagonal grid.

## Contribution

It applies automata theory and Karp's algorithm to coding theory problems, establishing a new lower bound for identifying code density on the hexagonal grid.

## Key findings

- New upper bound of approximately 0.4206 for the minimum density on the hexagonal grid.
- Improved the previous record of 3/7 (~0.4286) for identifying code density.
- Demonstrated the effectiveness of automata-based methods in coding theory.

## Abstract

We apply automata theory and Karp's minimum mean weight cycle algorithm to minimum density problems in coding theory. Using this method, we find the new upper bound $53/126 \approx 0.4206$ for the minimum density of an identifying code on the infinite hexagonal grid, down from the previous record of $3/7 \approx 0.4286$.

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Source: https://tomesphere.com/paper/2303.00557