The strategy that we have developed in this work is to combine BH with the quasi-random (QR) sampling approach. In our pure QR method, the conversion of each quasi-random vector into a trial crystal structure was followed by local energy minimization. Here, we replace the single local energy minimization of each QR trial structure by a BH trajectory. Our intention is that the quasi-random seeding of basin hopping simulations maintains some of the benefits of the low discrepancy approach, such as its uniform sampling of the configuration space of crystal packing, while benefiting from the efficiency of BH at moving towards low energy structures. The approach also maintains a certain level of parallelizability: each BH trajectory can be performed in parallel.