  The concept of paraunitary (PU) matrices arose in the early 1990s in the
study of multi-rate filter banks. So far, 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
$Z$-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
$Z$-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. Finally, we propose a unifying construction framework for optimal ZPU
matrices which includes existing PU matrices as a special case.
