Minimum densities for larger-radius locating-dominating codes in the triangular grid

Determine the minimum densities of $(r,\leq\ell)$-locating-dominating codes of types A and B in the infinite triangular grid for radii $r>1$, including the dependence on $r$ and $\ell$.

Background

The paper's main results concern the radius-one case r=1r=1 in the infinite triangular grid, where all values of ℓ\ell are resolved for both type-A and type-B codes. The authors explicitly do not extend these exact-density results to larger radii.

Thus, determining the corresponding minimum densities for r>1r>1 remains a concrete direction beyond the results established in the paper.

References

While we have focused on only the case with $r=1$, the cases with larger $r$ are left for future research.

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