# 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 $$. 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

$$b\left(\genfrac{}{}{0pt}{}{\kappa}{\tau}\right)=\lambda \left(\genfrac{}{}{0pt}{}{\nu}{\tau}\right).$$ |

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 |
---|---|

Canonical name | DifferenceSet |

Date of creation | 2013-03-22 16:50:04 |

Last modified on | 2013-03-22 16:50:04 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 05B10 |

Defines | non-trivial difference set |

Defines | planar difference set |