In combinatorics difference set of size of a group of order such that every non-identity is said to be cyclic, abelian, non-abelian is sometimes called planar or simple. is an abelian group can be written as a difference ways. The term "difference set" arises in this way.

Basic facts

• is called the order of the difference set.

• is also a difference set, and is called a translate in additive notation).

• -difference set.

forms a symmetric block design, called the development elements blocks acts as an automorphism group of the design. It is sharply transitive on both points and blocks.

• , then the difference set gives rise to a projective plane. 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.

Equivalent and isomorphic difference sets

are equivalent if there is a group isomorphism The two difference sets are isomorphic 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.

Multipliers

A multiplier is a group automorphism is called a numerical or Hall multiplier.

The multiplier conjecture predicts, in particular, that if p is a prime and not dividing v (the least common multiple of the orders is an integer coprime such that, for every prime p dividing m, there is an integer i. can be replaced by a weaker divisibility condition, but the full multiplier conjecture remains open.

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

Ryser and Lander conjectures

; consequently, the circulant Hadamard conjecture is a special case.

Lander's conjecture, proposed by Eric S. Lander, then the Sylow is not cyclic. Since every Sylow subgroup of a cyclic group is cyclic, Lander's conjecture implies Ryser's cyclic difference-set conjecture.

Lander's conjecture is unresolved in general. It has been proved when the order of the difference set is a power of a prime greater than 3, and for all difference sets in the McFarland, Spence, and Davis–Jedwab–Chen families. If the order is a power of 2 or 3 and the relevant Sylow subgroup is cyclic, then that subgroup must have order 2 or 3, respectively, and the parameters are restricted to a small number of exceptional forms. The conjecture is also known for all Hadamard difference sets of order at most 529.

, so Ryser's conjecture, and hence Lander's, predicts that no such difference set exists.

Parameters

Some important infinite parameter families, considered up to complementation, are:

• -difference set for some prime power. These are known as the classical parameters and there are many constructions of difference sets having these parameters.

• -difference set for some positive integer Difference sets with are called Paley-type difference sets.

• -difference set for some positive integer A difference set with these parameters is called a Hadamard difference set or Menon difference set. The existence of such a difference set in a cyclic group is the subject of Ryser's conjecture on circulant Hadamard matrices, a well-known open problem.

• and some positive integer Known as the McFarland parameters.

• -difference set for some positive integer Known as the Spence parameters.

• and some positive integer Difference sets with these parameters are called Davis–Jedwab–Chen difference sets.

Selected constructions

In many constructions of difference sets, the groups that are used are related to the additive and multiplicative groups of finite fields is the Galois field is the multiplicative group of non-zero elements.

• -difference set:

  1. be the set of all non-zero squares.

  -difference set:

  1. is the trace function

  Twin prime are both prime powers:

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

Xia, Zhou and Giannakis used difference sets to construct complex vector codebooks that attain the Welch lower bound on maximum cross-correlation amplitude in certain cases.

Generalisations

difference family ways.

of a difference family is a 2-design

See also

• Combinatorial design

• Ryser's conjecture on circulant Hadamard matrices

Notes

References

Further reading

• .