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

Basic facts

• A simple counting argument shows that there are exactly $k^{2}-k$ pairs of elements from $D$ that will yield nonidentity elements, so every difference set must satisfy the equation $k^{2}-k=(v-1)\lambda$.
• If $D$ is a difference set, and $g\in G$, then $gD=\{gd:d\in D\}$ is also a difference set, and is called a translate of $D$ ($D+g$ in additive notation).
• The complement of a $(v,k,\lambda )$-difference set is a $(v,v-k,v-2k+\lambda )$-difference set.
• The set of all translates of a difference set $D$ forms a symmetric block design. In such a design there are $v$ elements (usually called points) and $v$ blocks (subsets). Each block of the design consists of $k$ points, each point is contained in $k$ blocks. Any two blocks have exactly $\lambda$ elements in common and any two points are simultaneously contained in exactly $\lambda$ blocks. The group $G$ acts as an automorphism group of the design. It is sharply transitive on both points and blocks.
  • In particular, if $\lambda =1$, then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group $\mathbb {Z} /7\mathbb {Z}$ is the subset $\{1,2,4\}$. The translates of this difference set form the Fano plane.
• Since every difference set gives a symmetric design, the parameter set must satisfy the Bruck–Ryser–Chowla theorem.
• Not every symmetric design gives a difference set.

Multipliers

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

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 multipler). 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:

• $((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))$-difference set for some prime power $q$ and some positive integer $n$.
• $(4n-1,2n-1,n-1)$-difference set for some positive integer $n$.
• $(4n^{2},2n^{2}-n,n^{2}-n)$-difference set for some positive integer $n$.
• $(q^{n+1}(1+(q^{n+1}-1)/(q-1)),q^{n}(q^{n+1}-1)/(q-1),q^{n}(q^{n}-1)(q-1))$-difference set for some prime power $q$ and some positive integer $n$.
• $(3^{n+1}(3^{n+1}-1)/2,3^{n}(3^{n+1}+1)/2,3^{n}(3^{n}+1)/2)$-difference set for some positive integer $n$.
• $(4q^{2n}(q^{2n}-1)/(q-1),q^{2n-1}(1+2(q^{2n}-1)/(q+1)),q^{2n-1}(q^{2n-1}+1)(q-1)/(q+1))$-difference set for some prime power $q$ and some positive integer $n$.

Known difference sets

• Paley $(4n-1,2n-1,n-1)$-difference set:

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.

• Singer $((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))$-difference set:

Let $G={\rm {GF}}(q^{n+2})^{*}/{\rm {GF}}(q)^{*}$, where ${\rm {GF}}(q^{n+2})$ and ${\rm {GF}}(q)$ are Galois fields of orders $q^{n+2}$ and $q~$ respectively and ${\rm {GF}}(q^{n+2})^{*}$ and ${\rm {GF}}(q)^{*}$ are their respective multiplicative groups of non-zero elements. 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 ${\rm {Tr}}_{q^{n+2}/q}(x)=x+x^{q}+\cdots +x^{q^{n+1}}$.

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.

Generalisations

A $(v,k,\lambda ,s)$ difference family is a set of subsets $B=\{B_{1},...B_{s}\}$ of a group $G$ such that the order of $G$ is $v$, the size of $B_{i}$ is $k$ for all $i$, and every nonidentity element of $G$ can be expressed as a product $d_{1}d_{2}^{-1}$ of elements of $B_{i}$ for some $i$ (i.e. both $d_{1},d_{2}$ come from the same $B_{i}$) in exactly $\lambda$ ways.

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

See also

• Combinatorial design

References

• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge University Press, ISBN 0521333342
• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd Edition ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8 {{citation}}: |edition= has extra text (help)
• Storer, T. (1967). Cyclotomy and difference sets. Chicago: Markham Publishing Company. Zbl 0157.03301.
• van Lint, J.H.; Wilson, R.M. (1992), A Course in Combinatorics, Cambridge: Cambridge University Press, ISBN 0-521-42260-4
• Wallis, W.D. (1988). Combinatorial Designs. Marcel Dekker. ISBN 0-8247-7942-8. Zbl 0637.05004.
• Zwillinger, Daniel (2003). CRC Standard Mathematical Tables and Formulae. CRC Press. p. 246. ISBN 1-58488-291-3.
• Xia, Pengfei; Zhou, Shengli; Giannakis, Georgios B. (2005). "Achieving the Welch Bound with Difference Sets" (PDF). IEEE Transactions on Information Theory. 51 (5): 1900–1907. doi:10.1109/TIT.2005.846411. ISSN 0018-9448. Zbl 1237.94007..

Xia, Pengfei; Zhou, Shengli; Giannakis, Georgios B. (2006). "Correction to ``Achieving the Welch bound with difference sets"". IEEE Trans. Inf. Theory. 52 (7): 3359. Zbl 1237.94008.