Let G be a simple, undirected graph with vertex set V. For v is an element of V and r >= 1, we denote by B-G,B-r(v) the ball of radius r and centre v. A set C subset of V is said to be an r-identifying code in G if the sets B-G,B-r(V) boolean AND C, v is an element of V, are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free or r-identifiable, and in this case the smallest size of an r-identifying code in G is denoted by gamma(ID)(r)(G). We study the number of different optimal r-identifying codes C, i.e., such that vertical bar C vertical bar = gamma(ID)(r)(G), that a graph G can admit, and try to construct graphs having "many" such codes.