MacWilliams' Identity for metrics determined by directed graphs

Considering metrics based on finite directed graph, introduced by Etzion and Firer, we characterize the graphs such that every linear code admits a G-canonical decomposition. This decomposition will play an important role in this work, since it will be the main tool to give a sufficient condition for a finite directed graph to satisfy both the MacWilliams Identity and the MacWilliams Extension Property.