On identifying codes in lattices

Let G(V,E) be a simple, undirected graph. An identifying code on G is a vertex-subset C ⊆ V such that B(v) ∩ C is non-empty and distinct for each vertex v ∈ V , where B(v) is a ball about v. Motivated by applications to fault diagnosis in multiprocessor arrays, a number of researchers have considered the problem of constructing identifying codes of minimum density on various two-dimensional lattices, for instance, the square lattice. All currently known constructions of such codes are periodic, and none match the best known corresponding lower bounds on the code density. In this work, we give an explicit construction of a two-dimensional lattice G and identifying code C on G such that C is aperiodic and has optimal density δ, but every periodic identifying code on G has density greater than δ. In other words, we show that current constructive techniques are, in general, insufficient to achieve optimal code densities.