Generalization of Scarpis's theorem on Hadamard matrices

Abstract

A $\{1,-1\}$-matrix $H$ of order $m$ is a Hadamard matrix if $HH^T=mI_m$, where $T$ is the transposition operator and $I_m$ the identity matrix of order $m$. J. Hadamard published his paper on Hadamard matrices in 1893. Five years later, Scarpis showed how one can use a Hadamard matrix of order $n=1+p$, $p\equiv 3 \pmod{4}$ a prime, to construct a bigger Hadamard matrix of order $pn$. In this note we show that Scarpis's construction can be extended to the more general case where $p$ is replaced by a prime power $q$.