This paper presents an articulatory codebook construction method which gives a good space coverage with a limited number of points. For this aim, the articulatory space is decomposed into regions, represented by hypercubes, where the mapping is quasi-linear. The codebook, therefore, is a collection of hypercubes. The main quality of this codebook is that it respects the non-linear nature of the articulatory-to-acoustic mapping. We also present an inversion method using this hypercube codebook.