Design Theory  Part III
Org:
Andrea Burgess (University of New Brunswick),
Peter Danziger (Ryerson University) and
David Pike (Memorial University of Newfoundland)
[
PDF]
 HADI KHARIGHANI, University of Lethbridge
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 quasiunbiased weighing matrices with various parameters. \
This is joint work with Sho Suda.
 TRENT MARBACH, Nankai University
Balanced Equinsquares [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.