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{\large {\bf The Geometry of Frequency Squares}}
\\ [+.1in]
{\bf Dieter Jungnickel\\
University of Augsburg, Germany} \\[2mm]
(joint work with V.C.Mavron and T.P.McDonough)
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\noindent
\vspace{2mm}

The concept of a frequency square is a generalisation of that of a latin
square: A \emph{type} $F(n,\mu)$ \emph{frequency square} over a set 
$S$ of order
$m\geq2$ is an $n\times n$ matrix over $S$ such that each element of $S$
appears exactly $\mu$ times in every row and column, so $n=\mu m$. As
in the case of latin squares, there is a natural concept of
orthogonality: Two frequency squares $M,N$ of type $F(n,\mu)$ are 
\emph{orthogonal} if they
are over the same set $S$ and every element of $S^{2}$ occurs exactly $\mu
^{2}$ times among the pairs $(M_{ij},N_{ij})$, $1\leq i,j\leq n$.

  The maximum number of latin squares that are mutually
orthogonal to one another is bounded by a simple function of the
order of the squares, and sets of latin squares realising this bound
are called \emph{complete}. R. C. Bose established
that the existence of a complete set is equivalent to that of an
affine plane. Laywine and Mullen investigated this situation with the aim of
extending Bose's theorem to frequency squares. They established links between
affine 2-designs and complete sets of mutually orthogonal frequency squares
(MOFS). However, this approach does not give equivalence, as there are affine
2-designs which do not arise from complete sets of MOFS.

We shall provide a definitive answer to the problem of finding a geometric
interpretation of MOFS by showing that they are equivalent to a 
certain special type of nets
which we decided to call \emph{framed nets}. We use this result to 
give a new proof for the
bound on the maximum size of a set of MOFS by a counting argument 
which permits us to show
that the existence of a complete set (that is one realising the 
bound) of MOFS is equivalent
to the existence of a certain type of partial design (PBIBD). In the
case of latin squares, this PBIBD is an affine plane with two parallel classes
deleted. In consequence, we finally obtain the proper generalisation of Bose's
theorem which had proved elusive for such a long time. We also discuss examples
obtained from classical affine geometry and provide recursive construction
methods for (completely) framed nets, in this way unifying all known
construction methods for complete sets of MOFS.

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For background, see either of the following references:


A. Hedayat, N. J. A. Sloane and J. Stufken,
\textit{Orthogonal Arrays},
Springer, 1999.

C. F. Laywine and G. L. Mullen,
\textit{Discrete Mathematics using Latin Squares},
Wiley-Interscience, 1998.

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