Abelian difference set

Let $ G $ be a group of order $ v $ and $ D \subseteq G $ with $ | D | = k $. Then $ D $ is called a $ ( v,k, \lambda ) $- difference set of order $ n = k - \lambda $ in $ G $ if every element $ g \neq 1 $ in $ G $ has exactly $ \lambda $ different representations $ g = d \cdot d ^ {\prime - 1 } $ with $ d,d ^ \prime \in D $, see [a1]. For instance, $ \{ 1,2,4 \} $ is a $ ( 7,3,1 ) $- difference set in the cyclic group of order $ 7 $. If $ G $ is Abelian (cyclic, non-Abelian), the difference set is called Abelian (cyclic, non-Abelian). Two difference sets $ D _ {1} $ and $ D _ {2} $ in $ G $ are equivalent if there is a group automorphism $ \varphi $ such that $ \varphi ( D _ {1} ) = D _ {2} g $. The existence of a $ ( v,k, \lambda ) $- difference set is equivalent to the existence of a symmetric $ ( v,k, \lambda ) $- design with $ G $ acting as a regular automorphism group (cf. also Difference set). If two difference sets correspond to isomorphic designs, the difference sets are called isomorphic. Given certain parameters $ v $, $ k $ and $ \lambda $ and a group $ G $, the problem is to construct a difference set with those parameters or prove non-existence. To prove non-existence of Abelian difference sets, results from algebraic number theory are required: The existence of the difference set implies the existence of an algebraic integer of absolute value $ n $ in some cyclotomic field. In several cases one can prove that no such element exists, see [a5]. Another approach for non-existence results uses multipliers: A multiplier of an Abelian difference set in $ G $ is an automorphism $ \varphi $ of $ G $ such that $ \varphi ( D ) = Dg $. A statement that certain group automorphisms have to be multipliers of putative difference sets is called a multiplier theorem. It is known, for instance, that the mapping $ g \mapsto g ^ {t} $ is a multiplier of an Abelian difference set provided that: i) $ t $ divides the order $ n $; ii) $ t $ is relatively prime to $ v $; and iii) $ t > \lambda $. Several generalizations of this theorem are known, see [a1].

On the existence side, some families of Abelian difference sets are known, see [a3].

Examples.

The most popular examples are as follows.

Cyclic $ \left ( { \frac{q ^ {d + 1 } - 1 }{q - 1 } } , { \frac{q ^ {d} - 1 }{q - 1 } } , { \frac{q ^ {d - 1 } - 1 }{q - 1 } } \right ) $- difference sets, $ q $ a prime power. The classical construction of these difference sets (elements in the multiplicative group of $ { \mathop{\rm GF} } ( q ^ {d + 1 } ) $ whose trace is $ 0 $) corresponds to the classical point-hyperplane designs of a finite projective space. For non-equivalent cyclic examples with the same parameters, see [a5].

Quadratic residues in $ { \mathop{\rm GF} } ( q ) $, $ q \equiv 3 ( { \mathop{\rm mod} } 4 ) $( Paley difference sets). Some other cyclotomic classes yield difference sets too, see [a1].

$ ( 4m ^ {2} , 2m ^ {2} - m, m ^ {2} - m ) $- difference sets, $ m = 2 ^ {a} 3 ^ {b} u ^ {2} $, where $ u $ is a product of odd prime numbers (Hadamard difference sets, [a2]). If $ m = 2 ^ {a} $, it is known that an Abelian Hadamard difference set exists if and only if the exponent of $ G $ is at most $ 2 ^ {a + 2 } $, see [a4].

$ \left ( 4m ^ {2n } \cdot { \frac{m ^ {2n } - 1 }{m ^ {2} - 1 } } , m ^ {2n - 1 } \cdot \left ( { \frac{2 ( m ^ {2n } - 1 ) }{m + 1 } } + 1 \right ) , \right . $ $ \left . ( m ^ {2n } - m ^ {2n - 1 } ) \cdot { \frac{m ^ {2n - 1 } + 1 }{m + 1 } } \right ) $- difference sets, where $ m = q ^ {2} $( $ q $ an odd prime power) or $ m = 3 ^ {t} $ or $ m =2 $( generalized Hadamard difference sets, [a2]).

$ \left ( q ^ {d + 1 } \left ( 1 + { \frac{q ^ {d + 1 } - 1 }{q - 1 } } \right ) , q ^ {d} \cdot { \frac{q ^ {d + 1 } - 1 }{q - 1 } } , q ^ {d} \cdot { \frac{q ^ {d} - 1 }{q - 1 } } \right ) $- difference sets, $ q $ a prime power (McFarland difference sets).

References

[a1]	T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986)
[a2]	Y.Q. Chen, "On the existence of abelian Hadamard difference sets and generalized Hadamard difference sets" Finite Fields and Appl. (to appear)
[a3]	D. Jungnickel, A. Pott, "Difference sets: Abelian" Ch.J. Colbourn (ed.) J.H. Dinitz (ed.) , CRC Handbook of Combinatorial Designs , CRC (1996) pp. 297–307
[a4]	R.G. Kraemer, "Proof of a conjecture on Hadamard $2$-groups" J. Combinatorial Th. A , 63 (1993) pp. 1–10
[a5]	A. Pott, "Finite geometry and character theory" , Lecture Notes in Mathematics , 1601 , Springer (1995)