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.Note that it is enough to choose one such relation per conjugacy class of distinguished reflections, as all the corresponding braided reflections are conjugates in B (see [6]).The BMR freeness conjecture proposed by Broué, Malle and Rouquier [6] in 1998 states that the Hecke algebra H(W ) attached to W is a free R-module of rank equal to the order of W . After two decades, the BMR freeness conjecture is proven through the results of a number of authors. Thus, we have the following theorem.Theorem 2.3. The Hecke algebra H(W ) is a free R-module of rank |W |.The BMR freeness conjecture can be easily reduced to the case where W is irreducible. It is true for the (Iwahori-)Hecke algebra attached to any finite Coxeter group (see Lemma 4.4.3 of [11]). Ariki and Koike [2] proved it for the case of G(d, 1, n). Note that a basis for the Hecke algebra associated with G(d, 1, n) is also given in [4]. Ariki defined in [1] a Hecke algebra for G(de, e, n) by a presentation with generators and relations. He also proved that it is a free module of rank |G(de, e, n)|. The Hecke algebra defined by Ariki is isomorphic to the Hecke algebra defined by Broué, Malle, and Rouquier in [6] for G(de, e, n). The details why this is true can be found in Appendix A.2 of [18]. Hence one gets a proof of Theorem 2.3 for the general series of complex reflection groups.Concerning the exceptional complex reflection groups, Marin proved the conjecture for G 4 , G 25 , G 26 , and G 32 in [12] and [13]. Marin and Pfeiffer proved it for G 12 , G 22 , G 24 , G 27 , G 29 , G 31 , G 33 , and G 34 in [15]. In her PhD thesis and in the article that followed (see [7] and [8]), Chavli proved the validity of this conjecture for G 5 , G 6 , • • • , G 16 . Recently, Marin proved the conjecture for G 20 and G 21 (see [14]) and finally Tsushioka for G 17 , G 18 and G 19 (see [20]). Hence we obtain a proof of Theorem 2.3 for all the cases of irreducible complex reflection groups.