The number of completely different optimal identifying codes in the infinite square grid

Let G be a graph with vertex set V and edge set E. We call any subset C⊆V an identifying code if the sets I(v)={c∈C|{c,v}∈E or c=v}are distinct and non-empty for all vertices v∈V. We study identifying codes in the infinite square grid. The vertex set of this graph is Z2 and two vertices are connected by an edge if the Euclidean distance between these vertices is one. Ben-Haim & Litsyn have proved that the minimum density of identifying code in the infinite square grid is 720. Such codes are called optimal. We study the number of completely different optimal identifying codes in the infinite square grid. Two codes are called completely different if there exists an integer n such that no n×n-square of one code is equivalent with any n×n-square of the other code. In particular, we show that there are exactly two completely different optimal periodic codes and no optimal identifying code is completely different with both of these two periodic codes.