Systematic Constructions of Complementary Sets and Hadamard Matrices from Circulant Operator

A Hadamard matrix $H$ of order $n$ is a square matrix with entries $\pm 1$ satisfying $HH^T = nI_n$, where $I_n$ is the identity matrix of order $n$. A circulant Hadamard matrix is a Hadamard matrix whose rows are cyclic shifts of one another. This work establishes a unified algebraic framework that treats arbitrary Hadamard matrices as flexible seeds to systematically generate Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS) through algebraic transformations. In this paper, a systematic framework using cyclic operators is presented. First, circulant Hadamard matrices of order 4 are utilized recursively to propose binary CZCS of arbitrary lengths, achieving a maximum ZCZ ratio of 2/3, and binary GCS. Significantly, this framework is generalized to establish that by employing binary or complex Hadamard matrices of any order, binary or non-binary CZCSs of arbitrary lengths can be constructed with a ZCZ ratio of 1/2. Furthermore, to provide flexible user capacity, an alternative construction of binary GCS of all lengths and Hadamard matrices of order $2^{a+1} 10^b 26^c$ ($a, b, c \geq 0$) is proposed using circulant matrices and Golay complementary pairs (GCP). These constructions are further extended to form binary CCC with parameters $(2N, 2N, 2N)$, where $N=2^a 10^b 26^c$, and $(4n, 4n, 4n)$ for $n \geq 1$. Additionally, optimal binary $(8n, 8n, 8n, 4n)$-CZCSS and their complex versions with parameters $(4m, 4m, 4m, 2m)$ are proposed for $n, m \geq 1$. These results provide the first generalized framework for constructing optimal CZCSS from arbitrary Hadamard seeds. Finally, a theoretical relation between Hadamard matrices and GCSs is established, and fundamental properties of circulant matrices over aperiodic correlation functions are presented.