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

Geodesic Normal Forms and Hecke Algebras for the Complex Reflection Groups G(de,e,n)