Difference set

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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 difference set is a subset of size of a group of order such that every nonidentity element of can be expressed as a product of elements of in exactly ways. A difference set is said to be cyclic, abelian, non-abelian, etc., if the group has the corresponding property. A difference set with is sometimes called planar or simple.[1] If is an abelian group written in additive notation, the defining condition is that every nonzero element of can be written as a difference of elements of in exactly ways. The term "difference set" arises in this way.

Basic facts

Equivalent and isomorphic difference sets

Two difference sets in group and in group are equivalent if there is a group isomorphism between and such that for some . The two difference sets are isomorphic if the designs and 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]


A multiplier of a difference set in group is a group automorphism of such that for some . If is abelian and is the automorphism that maps , then is called a numerical or Hall multiplier.[7]

It has been conjectured that if p is a prime dividing and not dividing v, then the group automorphism defined by fixes some translate of D (this is equivalent to being a multiplier). It is known to be true for when 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 is a -difference set in an abelian group of exponent (the least common multiple of the orders of every element), let be an integer coprime to . If there exists a divisor of such that for every prime p dividing m, there exists an integer i with , 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 in an abelian group fixes a translate of , but it can also be shown that there is a translate of which is fixed by all numerical multipliers of .[9]


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, is the Galois field of order , where is a prime or prime power. The group under addition is denoted by , while is the multiplicative group of non-zero elements.

Let be a prime power. In the group , let be the set of all non-zero squares.
Let . Then the set is a -difference set, where is the trace function .
In the group , let [11]


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]


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.


A difference family is a set of subsets of a group such that the order of is , the size of is for all , and every nonidentity element of can be expressed as a product of elements of for some (i.e. both come from the same ) in exactly ways.

A difference set is a difference family with . The parameter equation above generalises to .[18] The development of a difference family is a 2-design. Every 2-design with a regular automorphism group is for some difference family .

See also


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