On a family of Abelian codes and their state complexities

We study Reed-Muller codes and “Berman” codes as Abelian codes. We show that the duals of Berman codes and Reed-Muller codes can be considered as belonging to the same family of Abelian codes. We also determine the minimum distance and state complexity of the duals of Berman codes. Each of the classical parameters generalizes that of Reed-Muller codes in the obvious way, but the state complexity does not. We conclude by comparing the asymptotic behavior of the state complexity of the duals of Berman codes with that of the obvious generalization of the state complexity of Reed-Muller codes