On Spatial Orders and Location Codes

Spatial orders such as the Morton (Z) order, U-order, or X-order have applications in matrix manipulation, graphic rendering and data encryption. It is shown that these spatial orders are single examples of entire classes of spatial orders which can be defined in arbitrary numbers of dimensions and base values. Secondly, an algorithm is proposed which can be used to transform between these spatial orders and Cartesian coordinates. It is shown that the efficiency of the algorithm improves with a larger base value. By choosing a base value that corresponds to the available memory page size, the computational effort required to perform operations such as matrix multiplication can be optimized.