Given an integer ≥1, a (1, ≤ )-identifying code in a digraph is a dominating subset C of vertices such that all distinct subsets of vertices of cardinality at most have distinct closed in-neighborhoods within C . In this paper, we prove that every line digraph of min- imum in-degree one does not admit a (1, ≤ )-identifying code for ≥3. Then we give a characterization so that a line digraph of a digraph different from a directed cycle of length 4 and minimum in-degree one admits a (1, ≤2)-identifying code. The identifying number of a digraph D , −→ γID (D ) , is the minimum size of all the identifying codes of D . We establish for digraphs without digons with both vertices of in-degree one that −→ γID (LD ) is lower bounded by the number of arcs of D minus the number of vertices with out-degree at least one. Then we show that −→ γID (LD ) attains the equality for a digraph having a 1- factor with minimum in-degree two and without digons with both vertices of in-degree two. We finish by giving an algorithm to construct identifying codes in oriented digraphs with minimum in-degree at least two and minimum out-degree at least one.