New Optimal $Z$-Complementary Code Sets Based on Generalized Paraunitary Matrices

The concept of paraunitary (PU) matrices arose in the early 1990 s in the study of multi-rate filter banks. These matrices have found wide applications in cryptography, digital signal processing, and wireless communications. Existing PU matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice. Motivated by this, for the first time, we introduce a novel concept, called <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula>-paraunitary (ZPU) matrix, whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value. We show that there exists an equivalence between a ZPU matrix and a <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula>-complementary code set when the latter is expressed as a matrix with polynomial entries. Furthermore, we investigate some important properties of ZPU matrices, which are useful for the extension of matrix sizes and sequence lengths. We propose a new construction framework for optimal ZPU matrices which includes existing PU matrices as special cases. Finally, we give the enumeration formula for the constructed <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula>-complementary sequences from the optimal ZPU matrices.