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The concept of paraunitary (PU) matrices arose in the early 1990s in thestudy of multi-rate filter banks. So far, these matrices have found wideapplications in cryptography, digital signal processing, and wirelesscommunications. Existing PU matrices are subject to certain constraints ontheir 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 ofpolynomials with identical degree not necessarily taking the maximum value. Weshow that there exists an equivalence between a ZPU matrix and a$Z$-complementary code set when the latter is expressed as a matrix withpolynomial entries. Furthermore, we investigate some important properties ofZPU matrices, which are useful for the extension of matrix sizes and sequencelengths. Finally, we propose a unifying construction framework for optimal ZPUmatrices which includes existing PU matrices as a special case.

New Optimal $Z$-Complementary Code Sets from Matrices of Polynomials