A computer-implemented method and system for generating large families of sequences with desirable properties for many applications, including communications and radar applications, applies constraints to a sequence in the Zak space, modulates the constrained sequence in the Zak space, and determines permutations of the modulated sequence in the Zak space. The constraints are associated with predetermined properties, including predetermined autocorrelation and cross-correlation properties. Other embodiments of the computer-implemented method and system transform an input sequence into a transformed sequence using the finite Zak transform and determine at least one other different sequence based on the transformed sequence. The at least one other different sequence can be determined by collecting a plurality of sequences that are finitely supported on an algebraic line in the Zak space and modulating and/or determining permutations of some or all of the sequences.