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)