Some Constructions for Amicable Orthogonal Designs

TL;DR

This paper investigates the existence of product designs for orthogonal designs, proves non-existence for certain orders, and constructs new classes of amicable orthogonal designs, including an infinite family and a specific large-order example.

Contribution

It establishes non-existence results for product designs of certain orders and types, and introduces new methods to construct classes of amicable orthogonal designs, including an infinite family.

Findings

No product design exists for orders other than 4, 8, 12 with specified types.

Constructed new classes of disjoint and full amicable orthogonal designs.

Developed an explicit example of a full amicable orthogonal design of order 512.

Abstract

Hadamard matrices, orthogonal designs and amicable orthogonal designs have a number of applications in coding theory, cryptography, wireless network communication and so on. Product designs were introduced by Robinson in order to construct orthogonal designs especially full orthogonal designs (no zero entries) with maximum number of variables for some orders. He constructed product designs of orders $4$, $8$ and $12$ and types $\big(1_{(3)}; 1_{(3)}; 1\big),$ $\big(1_{(3)}; 1_{(3)}; 5\big)$ and $\big(1_{(3)}; 1_{(3)}; 9\big)$, respectively. In this paper, we first show that there does not exist any product design of order $n\neq 4$, $8$, $12$ and type $\big(1_{(3)}; 1_{(3)}; n-3\big),$ where the notation $u_{(k)}$ is used to show that $u$ repeats $k$ times. Then, following the Holzmann and Kharaghani's methods, we construct some classes of disjoint and some classes of full amicable orthogonal designs, and we obtain an infinite class of full amicable orthogonal designs. Moreover, a full amicable orthogonal design of order $2^9$ and type $\big(2^6_{(8)}; 2^6_{(8)}\big)$ is constructed.