Purpose: This note discusses two level quasi-orthogonal matrices which were first highlighted by J. J. Sylvester; Hadamard
matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from
the unit disk. The goal of this note is to develop a theory of such matrices based on preliminary research results. Methods:
Our new regular Hadamard matrix constructed for order 196, suggests a source of ideas to construct regular Hadamard
matrices of orders n = 1 + p q = 1 + p (1 + 2m), where p, q are twin odd integer (q – p = 2); m = (q – 1)/2, prime, order of
inner blocks. Results: We present a new method aimed to give regular Hadamard matrix of order 196 and similar matrices.
Such kinds of regular Hadamard matrix of order 36 were done by Jennifer Seberry (1969), that inspired to find matrices of
orders 4k2, k integer, 36, 100, 196, …, 1444 and many others. We apply this result to the family of regular matrices obtaining
a new infinite family of Cretan matrices with orders 4t + 1, t an integer, 37, 101, 197, …, 1445, etc. Practical relevance: Web
addresses are given for other illustrations and other matrices with similar properties. Algorithms to construct regular matrices
have been implemented in developing software of the research program-complex. doi:10.15217/issn1684-8853.2015.1.2