A Proper $n$-Dimensional Orthogonal Design of Order $8$ on $8$ Indeterminates

Abstract

Let $x_1,x_2,\dots ,x_v$ be commuting indeterminates over the integers. We say an $v\times v\times v\dots \times v$ n-dimensional matrix is a proper $v$-dimensional orthogonal design of order $v$ and type $(s_1,s_2,\dots ,s_r)$ (written ${\mathrm{O}\mathrm{D}}^n(s_1,s_2,\dots ,s_r)$) on the indeterminates $x_1,x_2,\dots ,x_r$ if every 2-dimensional axis-normal submatrix is an $\mathrm{O}\mathrm{D}(s_1,s_2,\dots ,s_r)$ of order $v$ on the indeterminates $x_1,x_2,\dots ,x_r$. Constructions for proper ${\mathrm{O}\mathrm{D}}^n(1^2)$ of order 2 and ${\mathrm{O}\mathrm{D}}^n(1^4)$ of order 4 are given in J. Seberry (1980) and J. Hammer and J. Seberry (1979, 1981a), respectively. This paper contains simple constructions for proper ${\mathrm{O}\mathrm{D}}^n(1^2)$, ${\mathrm{O}\mathrm{D}}^n(1^4)$, and ${\mathrm{O}\mathrm{D}}^n(1^8)$ of orders 2, 4, and 8, respectively. Prior to this paper no proper higher dimensional OD on more than 4 indeterminates was known.