Signed group orthogonal designs and their applications

Craigen introduced and studied {\it signed group Hadamard matrices} extensively in \cite{Craigenthesis, Craigen}. Livinskyi \cite{Ivan}, following Craigen's lead, studied and provided a better estimate for the asymptotic existence of signed group Hadamard matrices and consequently improved the asymptotic existence of Hadamard matrices. In this paper, we introduce and study signed group orthogonal designs. The main results include a method for finding signed group orthogonal designs for any $k$-tuple of positive integer and then an application to obtain orthogonal designs from signed group orthogonal designs, namely, for any $k$-tuple $\big(u_1, u_2, ..., u_{k}\big)$ of positive integers, we show that there is an integer $N=N(u_1, u_2, ..., u_k)$ such that for each $n\ge N$, a full orthogonal design (no zero entries) of type $\big(2^nu_1,2^nu_2,...,2^nu_{k}\big)$ exists . This is an alternative approach to the results obtained in \cite{EK}.