Signed group orthogonal designs and their applications

TL;DR

This paper introduces signed group orthogonal designs, providing a method to construct them for any positive integer tuple and applying these to generate orthogonal designs with specific parameters, advancing the theory of orthogonal matrices.

Contribution

It develops a new approach for constructing signed group orthogonal designs for any positive integer tuple and applies this to produce orthogonal designs with specified types.

Findings

Established a method for constructing signed group orthogonal designs for any k-tuple of positive integers.

Proved the existence of full orthogonal designs of certain types for sufficiently large n.

Provided an alternative approach to existing results on orthogonal designs.

Abstract

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}.