Locating-dominating codes in the infinite hexagonal grid

Determine the minimum densities of locating-dominating codes of types A and B in the infinite hexagonal grid for general parameters $r$ and $\ell$, extending beyond the currently known usual location-domination case and identifying-code-based upper bounds.

Background

The paper establishes exact densities for the infinite triangular grid and notes that the corresponding theory for the infinite hexagonal grid is largely undeveloped. For the hexagonal grid, the literature contains the usual locating-domination result for ℓ=1\ell=1 and results on (1,≤ℓ)(1,\leq\ell)-identifying codes that provide upper bounds for type-A and type-B locating-dominating codes.

Consequently, exact minimum densities for the broader family of LDA/LDB codes in the infinite hexagonal grid remain unresolved.

References

Yet another future research direction is considering LDA/LDB codes in the infinite hexagonal grid. There are currently no articles on the topic beyond the works of usual location-domination (giving the density for $\ell=1$) and works considering $(1,\leq\ell)$-identifying codes which provide upper bounds for the minimum density of LDA/LDB codes.

— Optimal $(1,\le \ell)$-locating-dominating codes in infinite triangular grid  (2610.03227 - Das et al., 2 Oct 2026) in Section Conclusion