is said to be cyclic, abelian, non-abelian, etc., if the group
is sometimes called planar or simple.
is an abelian group written in additive notation, the defining condition is that every non-zero element of
The term "difference set" arises in this way.
are equivalent if there is a group isomorphism
The two difference sets are isomorphic if the 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.
is called a numerical or Hall multiplier.
and not dividing v, then the group automorphism defined by
fixes some translate of D (this is equivalent to being a multiplier).
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
-difference set in an abelian group
(the least common multiple of the orders of every element), let
such that for every prime p dividing m, there exists an integer i with
[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
[9] The known difference sets or their complements have one of the following parameter sets:[10] 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.
The group under addition is denoted by
is the multiplicative group of non-zero elements.
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.
difference family is a set of subsets
Every 2-design with a regular automorphism group is