A New Method of Constructing Hadamard Matrices, Circulant Hadamard Matrices, CZCS, GCS, CCC, and CZCSS

A Hadamard matrix H is a square matrix of order n with entries ± 1, such that HH⁼nIₙ, where Iₙ is an identity matrix of order n. A circulant Hadamard matrix H is a Hadamard matrix that has rows of entries in cyclic order. There exist only $8$ circulant Hadamard matrices of order 4, and here, we provide a novel construction of all such $8$ circulant Hadamard matrices using a linear operator and generalized Boolean function (GBF). The constructed circulant Hadamard matrices are used recursively to construct a binary cross Z-complementary set (CZCS) of all lengths with an even phase, a binary Golay complementary set (GCS) of all lengths, and Hadamard matrices of order 2ⁿ⁺², where n≥1. The construction of a binary CZCS covering all lengths was not available before. We also propose an alternative, lower-complexity construction of binary GCSs of all lengths and Hadamard matrices of order 2ᵃ⁺¹10ᵇ26ᶜ using circulant matrices, where a,b,c ≥ 0. The proposed binary GCS covers all lengths with a flexible flock size. The constructions of GCS are further extended to form binary complete complementary code (CCC) of the parameter $(2N,2N,2N)-CCC$ where N=2ᵃ10ᵇ26ᶜ, a,b,c ≥ 0. The constructed binary CCC provides a flexible flock size. The construction of CZCS is further extended to form a binary optimal cross-Z complementary sequence set (CZCSS) of the parameter (2ⁿ⁺², 2ⁿ⁺², 2ⁿ⁺², 2ⁿ⁺¹)-CZCSS, where n≥1. Finally, we provide a relation between Hadamard matrices and GCS, which enables the study of the Hadamard conjecture in a new direction. We also provided a few properties of circulant matrices over aperiodic cross-correlation (ACCF) and aperiodic auto-correlation (AACF), which are used to prove the theorems. All proposed constructions are novel, and their parameters are compared with the existing state-of-the-art.