On the injectivity of modular mappings

Affine space-time mappings have been extensively studied for systolic array design and parallelizing compilation. However, there are practical important cases that require other types of transformations. This paper considers so-called modular mappings described by linear transformations modulo a constant vector. Sufficient conditions for these mappings to be one-to-one are investigated for rectangular domains of arbitrary dimensions. It is shown that a sufficient condition for a modular mapping to be one-to-one is that its (n×n) coefficient matrix T has entries t
_ii
=±1 and t
_ij
=0 for i