# Difference set

For the set of elements in one set but not another, see relative complement. For the set of differences of pairs of elements, see Minkowski difference.

## Equivalent and isomorphic difference sets

Equivalent difference sets are isomorphic, but there exist examples of isomorphic difference sets which are not equivalent. In the cyclic difference set case, all known isomorphic difference sets are equivalent.

## Multipliers

It has been conjectured that if p is a prime dividing $k-\lambda$ and not dividing v, then the group automorphism defined by $g\mapsto g^{p}$ fixes some translate of D (this is equivalent to being a multiplier). It is known to be true for $p>\lambda$ when $G$ is an abelian group, and this is known as the First Multiplier Theorem. A more general known result, the Second Multiplier Theorem, says that if $D$ is a $(v,k,\lambda )$ -difference set in an abelian group $G$ of exponent $v^{*}$ (the least common multiple of the orders of every element), let $t$ be an integer coprime to $v$ . If there exists a divisor $m>\lambda$ of $k-\lambda$ such that for every prime p dividing m, there exists an integer i with $t\equiv p^{i}\ {\pmod {v^{*}}}$ , then t is a numerical divisor.

For example, 2 is a multiplier of the (7,3,1)-difference set mentioned above.

It has been mentioned that a numerical multiplier of a difference set $D$ in an abelian group $G$ fixes a translate of $D$ , but it can also be shown that there is a translate of $D$ which is fixed by all numerical multipliers of $D$ .

## Parameters

The known difference sets or their complements have one of the following parameter sets:

## Known difference sets

In many constructions of difference sets the groups that are used are related to the additive and multiplicative groups of finite fields. The notation used to denote these fields differs according to discipline. In this section, ${\rm {GF}}(q)$ is the Galois field of order $q$ , where $q$ is a prime or prime power. The group under addition is denoted by $G=({\rm {GF}}(q),+)$ , while ${\rm {GF}}(q)^{*}$ is the multiplicative group of non-zero elements.

Let $q=4n-1$ be a prime power. In the group $G=({\rm {GF}}(q),+)$ , let $D$ be the set of all non-zero squares.
Let $G={\rm {GF}}(q^{n+2})^{*}/{\rm {GF}}(q)^{*}$ . Then the set $D=\{x\in G~|~{\rm {Tr}}_{q^{n+2}/q}(x)=0\}$ is a $((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))$ -difference set, where ${\rm {Tr}}_{q^{n+2}/q}:{\rm {GF}}(q^{n+2})\rightarrow {\rm {GF}}(q)$ is the trace function ${\rm {Tr}}_{q^{n+2}/q}(x)=x+x^{q}+\cdots +x^{q^{n+1}}$ .
In the group $G=({\rm {GF}}(q),+)\oplus ({\rm {GF}}(q+2),+)$ , let $D=\{(x,y)\colon y=0{\text{ or }}x{\text{ and }}y{\text{ are non-zero and both are squares or both are non-squares}}\}.$ ## History

The systematic use of cyclic difference sets and methods for the construction of symmetric block designs dates back to R. C. Bose and a seminal paper of his in 1939. However, various examples appeared earlier than this, such as the "Paley Difference Sets" which date back to 1933. The generalization of the cyclic difference set concept to more general groups is due to R.H. Bruck in 1955. Multipliers were introduced by Marshall Hall Jr. in 1947.

## Application

It is found by Xia, Zhou and Giannakis that, difference sets can be used to construct a complex vector codebook that achieves the difficult Welch bound on maximum cross correlation amplitude. The so-constructed codebook also forms the so-called Grassmannian manifold.