Abstract. Definitions for the uniform representation of d-dimensional matrices serially in Morton-order (or Z-order) support both their use with cartesian indices, and their divide-and-conquer manipulation as quaternary trees. In the latter case, d-dimensional arrays are accessed as 2 d-ary trees. This data structure is important because, at once, it relaxes serious problems of locality and latency, and the tree helps schedule multiprocessing. It enables algorithms that avoid cache misses and page faults at all levels in hierarchical memory, independently of a specific runtime environment. This paper gathers the properties of Morton order and its mappings to other indexings, and outlines for compiler support of it. Statistics elsewhere show that the new ordering and block algorithms achieve high flop rates and, indirectly, parallelism without any low-level tuning.

The Opie Project aims to develop a compiler to transform C codes written for row-major matrix representation into equivalentcodes for Morton-order matrix representation, and to apply its techniques to other languages. Accepting a possible reduction in performance weseek to compile libraries of usable code to support future developmentofnew algorithms better suited to Morton-ordered matrices.