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This article investigates the combinatorial properties of difference sets within the cyclic group Zn, specifically in the context of circulant partial Hadamard matrices (CPHMs). We examine the structural characteristics and establish relationships between difference sets associated with 2-H(m×n) and 0-H(m×n) matrices. Our results provide insights into the interplay between these matrix classes and their corresponding difference sets, contributing to the broader understanding of their applications in combinatorial design theory.

In this paper, some properties of circulant partial Hadamard matrices of the form \\(4-H(k\\times 2k)\\) have been obtained together with a method of construction with the help of Toeplitz matrices.

This article deals with some special cases of Williamson Hadamard matrices, which are generated by block symmetric circulant matrices. In these cases, the patterns of the obtained examples have been analyzed for insight into the nature of the Williamson matrices.

This paper provides some new results on \\(r\\)-row-regular circular partial Hadamard matrices of order \\((k\\times 2k)\\), and also discusses the possible linear relationship between \\(r\\) and \\(k\\). Furthermore, a method of constructing such a matrix is given.