New Construction of Complementary Sequence (or Array) Sets and Complete Complementary Codes (II)

Previously, we have presented a framework to use the para-unitary (PU) matrix-based approach for constructing new complementary sequence set (CSS), complete complementary code (CCC), complementary sequence array (CSA), and complete complementary array (CCA). In this paper, we introduce a new class of delay matrices for the PU construction. In this way, generalized Boolean functions (GBF) derived from PU matrix can be represented by an array of size $2\times 2 \times \cdots \times 2$. In addition, we introduce a new method to construct PU matrices using block matrices. With these two new ingredients, our new framework can construct an extremely large number of new CSA, CCA, CSS and CCC, and their respective GBFs can be also determined recursively. Furthermore, we can show that the known constructions of CSSs, proposed by Paterson and Schmidt respectively, the known CCCs based on Reed-muller codes are all special cases of this new framework. In addition, we are able to explain the bound of PMEPR of the sequences in the part of the open question, proposed in 2000 by Paterson.Comment: This paper and another is merged together. And the merged paper is onlin

New Optimal ZZ-Complementary Code Sets from Matrices of Polynomials

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