Hadamard matrices constructed from supplementary difference sets in the class ℋ︁<sub>1</sub>

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<jats:title>Abstract</jats:title><jats:p>In this article we give the definition of the class ℋ︁<jats:sub>1</jats:sub> and prove: (1) ℋ︁<jats:sub>1</jats:sub>(<jats:italic>v</jats:italic>) ≠ ϕ for <jats:italic>v</jats:italic> ∈ 𝒩 = 𝒩<jats:sub>1</jats:sub> ∪ 𝒩<jats:sub>2</jats:sub> ∪ 𝒩<jats:sub>3</jats:sub> where <jats:disp-formula> </jats:disp-formula> <jats:disp-formula> </jats:disp-formula> <jats:disp-formula> </jats:disp-formula></jats:p><jats:p>(2) there exists 2 − {2<jats:italic>q<jats:sup>2</jats:sup>; q<jats:sup>2</jats:sup> ± q, q<jats:sup>2</jats:sup>;q<jats:sup>2</jats:sup> ± q</jats:italic>} supplementary difference sets for <jats:italic>q</jats:italic><jats:sup>2</jats:sup> ∈ 𝒩; (3) there exists an Hadamard matrix of order 4<jats:italic>v</jats:italic> for <jats:italic>v</jats:italic> ∈ 𝒩; (4) if <jats:italic>t</jats:italic> is an order of T‐matrices, there exists an Hadamard matrix of order 4<jats:italic>tv</jats:italic> for <jats:italic>v</jats:italic> ∈ 𝒩. © 1994 John Wiley & Sons, Inc.</jats:p>

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