  We establish geodesic normal forms for the general series of complex
reflection groups G(de,e,n) by using the presentations of Corran-Picantin and
Corran-Lee-Lee of G(e,e,n) and G(de,e,n) for d > 1, respectively. This requires
the elaboration of a combinatorial technique in order to explicitly determine
minimal word representatives of the elements of G(de,e,n). Using these geodesic
normal forms, we construct natural bases for the Hecke algebras associated with
the complex reflection groups G(e,e,n) and G(d,1,n). As an application, we
obtain a new proof of the BMR freeness conjecture for these groups.
