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.

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 $3/7 \approx 0.4286$.