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 $Z$-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. 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 $Z$-complementary sequences from the optimal ZPU matrices.

中文翻译：

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基于广义准矩阵的新最优 Z 互补码集

超幺正 (PU) 矩阵的概念出现在 1990 年代初期的多速率滤波器组研究中。这些矩阵在密码学、数字信号处理和无线通信中得到了广泛的应用。现有 PU 矩阵的存在受到某些限制，因此在实践中无法保证其可用性。受此启发，我们首次引入了一个新颖的概念，称为$Z$-paraunitary (ZPU) 矩阵，其正交性定义在具有相同次数的多项式矩阵上，不一定取最大值。我们证明了 ZPU 矩阵和$Z$-当后者表示为具有多项式项的矩阵时的互补代码集。此外，我们研究了 ZPU 矩阵的一些重要属性，这些属性对于矩阵大小和序列长度的扩展很有用。我们为最优 ZPU 矩阵提出了一个新的构建框架，其中包括现有的 PU 矩阵作为特殊情况。最后，我们给出构造的枚举公式$Z$- 来自最佳 ZPU 矩阵的互补序列。