05B10-DifferenceSet.tex

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\pmtitle{difference set}
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\pmdefines{planar difference set}

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\begin{document}
\textbf{Definition}.  Let $A$ be a finite abelian group of order $n$.  A subset $D$ of $A$ is said to be a \emph{difference set} (in $A$) if there is a positive integer $m$ such that every non-zero element of $A$ can be expressed as the difference of elements of $D$ in exactly $m$ ways.  

If $D$ has $d$ elements, then we have the equation $$m(n-1)=d(d-1).$$  In the equation, we are counting the number of pairs of distinct elements of $D$.  On the left hand side, we are counting it by noting that there are $m(n-1)$ pairs of elements of $D$ such that their difference is non-zero.  On the right hand side, we first count the number of elements in $D^2$, which is $d^2$, then subtracted by $d$, since there are $d$ pairs of $(x,y)\in D^2$ such that $x=y$.

A difference set with parameters $n,m,d$ defined above is also called a $(n,d,m)$-difference set.  A difference set is said to be \emph{non-trivial} if $1<d <n-1$.  A difference set is said to be \emph{planar} if $m=1$.

\textbf{Difference sets versus square designs}.  Recall that a square design is a $\tau$-$(\nu,\kappa,\lambda)$-\PMlinkname{design}{Design} where $\tau=2$ and the number $\nu$ of points is the same as the number $b$ of blocks.  In a general design, $b$ is related to the other numbers by the equation $$b\binom{\kappa}{\tau}=\lambda\binom{\nu}{\tau}.$$  So in a square design, the equation reduces to $b\kappa(\kappa-1)=\lambda\nu(\nu-1)$, or $$\lambda(\nu-1)=\kappa(\kappa-1),$$
which is identical to the equation above for the difference set.  A square design with parameters $\lambda,\nu,\kappa$ is called a square $(\nu,\kappa,\lambda)$-design.  

One can show that a subset $D$ of an abelian group $A$ is an $(n,d,m)$-difference set iff it is a square $(n,d,m)$-design where $A$ is the set of points and $\lbrace D+a\mid a\in A\rbrace$ is the set of blocks.
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