On locating-dominating codes for locating large numbers of vertices in the infinite king grid

Assume that G = (V, E) is an undirected graph with vertex set V and edge set E. The ball Br(v) denotes the vertices within graphical distance r from v. A subset C ⊆ V is called an (r,≤ l)-locating-dominating code of type B if the sets Ir(F ) = ⋃ v∈F (Br(v)∩C) are distinct for all subsets F ⊆ V \C with at most l vertices. A subset C ⊆ V is an (r,≤ l)-locatingdominating code of type A if sets Ir(F1) and Ir(F2) are distinct for all subsets F1, F2 ⊆ V where F1 = F2, F1 ∩ C = F2 ∩ C and |F1|, |F2| ≤ l. We study (r,≤ l)-locating-dominating codes in the infinite king grid when r ≥ 1 and l ≥ 3. The infinite king grid is the graph with vertex set Z and edge set {{(x1, y1), (x2, y2)} | |x1 − x2| ≤ 1, |y1 − y2| ≤ 1}.