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Homedifference set

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# 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 $1<d<n-1$. 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 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.

## Mathematics Subject Classification

05B10*no label found*

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