Generalization of Scarpis's theorem on Hadamard matrices

TL;DR

This paper extends Scarpis's method for constructing larger Hadamard matrices from prime-based matrices to those based on prime powers, broadening the scope of Hadamard matrix generation.

Contribution

It generalizes Scarpis's theorem, allowing the construction of larger Hadamard matrices using prime power orders instead of just primes.

Findings

Extended the construction to prime power orders

Broadened the class of Hadamard matrices that can be generated

Provided a new method for Hadamard matrix construction

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 $q$.