Hadamard matrices generated by an adaptation of generalized quaternion type array (Q1074587)

Let L denote the ring of generalized quaternions. Then the so-called right regular representation matrix of a particular element of L gives rise to a Hadamard matrix of order 2n of generalized quaternion type (GQT). The author establishes the existence of a Hadamard matrix of order \(4(n+1)\) which is an adaptation of GQT arrays. Similar theorems have been obtained by \textit{E. Spence} [J. Comb. Theory, Ser. A 19, 287-300 (1975; Zbl 0316.05018)] and \textit{A. L. Whiteman} [J. Number Theory 8, 1-11 (1976; Zbl 0329.05014)] by means of Goethals-Seidel arrays. The main result of the paper under review reads as follows. If \(q\equiv 1 (mod 4)\) and q-2 are prime numbers, then there exists a Hadamard matrix of order 4q. As an important tool of the proof, relative Gauß sums over a finite field are studied.