Design Theory - Part III
Org: Andrea Burgess (University of New Brunswick), Peter Danziger (Ryerson University) and David Pike (Memorial University of Newfoundland)
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Unbiased Orthogonal Designs  [PDF]

Let $D_1,D_2$ be orthogonal designs of order $n$ and type $(s_1,\ldots,s_u)$ in variables $x_1,\ldots,x_u$. $D_1$ and $D_2$ are unbiased with parameter $\alpha$ if $\alpha$ is a positive real number and there exists a $(0,1,-1)$-matrix $W$ such that \begin{align*} D_1D_2^\top=\frac{s_1x_1^2+\cdots+s_ux _u^2}{\sqrt{\alpha}}W. \end{align*}

The study of unbiased orthogonal designs provides a unified approach to the study of a variety of unbiased matrices.

Upper bounds, an asymptotic existence result, and some methods of construction will be presented. The application includes the proof of the existence of mutually quasi-unbiased weighing matrices with various parameters. \

This is joint work with Sho Suda.

TRENT MARBACH, Nankai University
Balanced Equi-n-squares  [PDF]

In this presentation, we present recent work undertaken to understand $d$-balanced equi- $n$-squares. With the requirement that $d$ is a divisor of $n$, these structures are $n\times n$ matrices containing symbols from $\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. There are connections with Latin square of order $n$ that decompose into $d \times(n/d)$ subrectangles, which we exploit to construct $d$-balanced equi-$n$-squares. We also show connections with $\alpha$-labellings of graphs, which enables us to both construct new $d$-balanced equi-$n$-squares and construct new $\alpha$-labellings of graphs.

BRETT STEVENS, Carleton University
Affine planes with ovals for blocks  [PDF]

A beautiful theorem states that the reverse of a line in the Singer Cycle presentation of a projective plane is an oval. This implies that for every Desarguesian projective plane there is a companion plane all of whose blocks are ovals in the first. This fact has been exploited to construct a family of very efficient strength 3 covering arrays. We show that there exist pairs of Desarguesian affine planes whose blocks are ovals in the other plane for any order a power of 2. These can be used to construct efficient covering arrays.

DOUG STINSON, University of Waterloo
Constructions of optimal orthogonal arrays with repeated rows  [PDF]

We construct orthogonal arrays OA$_{\lambda}(k,n)$ (of strength two) having a row that is repeated $m$ times, where the ratio $m/\lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k \geq n + 1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which gcd$(m, \lambda) = 1$. We construct a basic OA with $n = 2$ and $k = 4t+1$, provided that a Hadamard matrix of order $8t+4$ exists.

This is joint work with Charlie Colbourn and Shannon Veitch.