# 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.

In combinatorics, a ${\displaystyle (v,k,\lambda )}$ difference set is a subset ${\displaystyle D}$ of size ${\displaystyle k}$ of a group ${\displaystyle G}$ of order ${\displaystyle v}$ such that every nonidentity element of ${\displaystyle G}$ can be expressed as a product ${\displaystyle d_{1}d_{2}^{-1}}$ of elements of ${\displaystyle D}$ in exactly ${\displaystyle \lambda }$ ways. A difference set ${\displaystyle D}$ is said to be cyclic, abelian, non-abelian, etc., if the group ${\displaystyle G}$ has the corresponding property. A difference set with ${\displaystyle \lambda =1}$ is sometimes called planar or simple.[1] If ${\displaystyle G}$ is an abelian group written in additive notation, the defining condition is that every nonzero element of ${\displaystyle G}$ can be written as a difference of elements of ${\displaystyle D}$ in exactly ${\displaystyle \lambda }$ ways. The term "difference set" arises in this way.

## Equivalent and isomorphic difference sets

Two difference sets ${\displaystyle D_{1}}$ in group ${\displaystyle G_{1}}$ and ${\displaystyle D_{2}}$ in group ${\displaystyle G_{2}}$ are equivalent if there is a group isomorphism ${\displaystyle \psi }$ between ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ such that ${\displaystyle D_{1}^{\psi }=\{d^{\psi }\colon d\in D_{1}\}=gD_{2}}$ for some ${\displaystyle g\in G_{2}}$. The two difference sets are isomorphic if the designs ${\displaystyle dev(D_{1})}$ and ${\displaystyle dev(D_{2})}$ are isomorphic as block designs.

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.[6]

## Multipliers

A multiplier of a difference set ${\displaystyle D}$ in group ${\displaystyle G}$ is a group automorphism ${\displaystyle \phi }$ of ${\displaystyle G}$ such that ${\displaystyle D^{\phi }=gD}$ for some ${\displaystyle g\in G}$. If ${\displaystyle G}$ is abelian and ${\displaystyle \phi }$ is the automorphism that maps ${\displaystyle h\mapsto h^{t}}$, then ${\displaystyle t}$ is called a numerical or Hall multiplier.[7]

It has been conjectured that if p is a prime dividing ${\displaystyle k-\lambda }$ and not dividing v, then the group automorphism defined by ${\displaystyle g\mapsto g^{p}}$ fixes some translate of D (this is equivalent to being a multiplier). It is known to be true for ${\displaystyle p>\lambda }$ when ${\displaystyle G}$ 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 ${\displaystyle D}$ is a ${\displaystyle (v,k,\lambda )}$-difference set in an abelian group ${\displaystyle G}$ of exponent ${\displaystyle v^{*}}$ (the least common multiple of the orders of every element), let ${\displaystyle t}$ be an integer coprime to ${\displaystyle v}$. If there exists a divisor ${\displaystyle m>\lambda }$ of ${\displaystyle k-\lambda }$ such that for every prime p dividing m, there exists an integer i with ${\displaystyle t\equiv p^{i}\ {\pmod {v^{*}}}}$, then t is a numerical divisor.[8]

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 ${\displaystyle D}$ in an abelian group ${\displaystyle G}$ fixes a translate of ${\displaystyle D}$, but it can also be shown that there is a translate of ${\displaystyle D}$ which is fixed by all numerical multipliers of ${\displaystyle D}$.[9]

## Parameters

The known difference sets or their complements have one of the following parameter sets:[10]

## 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, ${\displaystyle {\rm {GF}}(q)}$ is the Galois field of order ${\displaystyle q}$, where ${\displaystyle q}$ is a prime or prime power. The group under addition is denoted by ${\displaystyle G=({\rm {GF}}(q),+)}$, while ${\displaystyle {\rm {GF}}(q)^{*}}$ is the multiplicative group of non-zero elements.

Let ${\displaystyle q=4n-1}$ be a prime power. In the group ${\displaystyle G=({\rm {GF}}(q),+)}$, let ${\displaystyle D}$ be the set of all non-zero squares.
Let ${\displaystyle G={\rm {GF}}(q^{n+2})^{*}/{\rm {GF}}(q)^{*}}$. Then the set ${\displaystyle D=\{x\in G~|~{\rm {Tr}}_{q^{n+2}/q}(x)=0\}}$ is a ${\displaystyle ((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))}$-difference set, where ${\displaystyle {\rm {Tr}}_{q^{n+2}/q}:{\rm {GF}}(q^{n+2})\rightarrow {\rm {GF}}(q)}$ is the trace function ${\displaystyle {\rm {Tr}}_{q^{n+2}/q}(x)=x+x^{q}+\cdots +x^{q^{n+1}}}$.
In the group ${\displaystyle G=({\rm {GF}}(q),+)\oplus ({\rm {GF}}(q+2),+)}$, let ${\displaystyle D=\{(x,y)\colon y=0{\text{ or }}x{\text{ and }}y{\text{ are non-zero and both are squares or both are non-squares}}\}.}$[11]

## 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.[12] However, various examples appeared earlier than this, such as the "Paley Difference Sets" which date back to 1933.[13] The generalization of the cyclic difference set concept to more general groups is due to R.H. Bruck[14] in 1955.[15] Multipliers were introduced by Marshall Hall Jr.[16] in 1947.[17]

## 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.

## Generalisations

A difference set is a difference family with ${\displaystyle s=1}$. The parameter equation above generalises to ${\displaystyle s(k^{2}-k)=(v-1)\lambda }$.[18] The development ${\displaystyle dev(B)=\{B_{i}+g:i=1,...,s,g\in G\}}$ of a difference family is a 2-design. Every 2-design with a regular automorphism group is ${\displaystyle dev(B)}$ for some difference family ${\displaystyle B}$.

## Notes

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## References

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