# difference set

Definition. Let $A$ be a finite abelian group of order $n$. A subset $D$ of $A$ is said to be a 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 non-trivial if $1. A difference set is said to be planar if $m=1$.

Difference sets versus square designs. Recall that a square design is a $\tau$-$(\nu,\kappa,\lambda)$-design (http://planetmath.org/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 $\{D+a\mid a\in A\}$ is the set of blocks.

Title difference set DifferenceSet 2013-03-22 16:50:04 2013-03-22 16:50:04 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 05B10 non-trivial difference set planar difference set