Some results on skew Hadamard difference sets

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v1 and v2 with |v1 − v2| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 35 or 37, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 39. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 311.

[1]  A. Brouwer,et al.  On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs , 1992 .

[2]  T. Storer.  Cyclotomy and difference sets , 1967 .

[3]  Mikhail E. Muzychuk,et al.  Strongly regular Cayley graphs over the group Zpn sumset Zpn , 2005, Discret. Math..

[4]  P. K. Menon.  On difference sets whose parameters satisfy a certain relation , 1962 .

[5]  Qing Xiang,et al.  Pseudo-Paley graphs and skew Hadamard difference sets from presemifields , 2007, Des. Codes Cryptogr..

[6]  L. D. Baumert.  Cyclic Difference Sets , 1971 .

[7]  Qing Xiang,et al.  An exponent bound on skew Hadamard abelian difference sets , 1994, Des. Codes Cryptogr..

[8]  Siu Lun Ma,et al.  Strongly regular Cayley graphs with l−m=−1 , 1994 .

[9]  Siu Lun Ma,et al.  Polynomial addition sets and symmetric difference sets , 1990 .

[10]  E. C Johnsen,et al.  Skew-hadamard abelian group difference sets , 1966 .

[11]  ROBERT E. KIBLER,et al.  A Summary of Noncyclic Difference Sets, k<20 , 1978, J. Comb. Theory, Ser. A.

[12]  Cunsheng Ding,et al.  Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3, 32h+1) , 2007, J. Comb. Theory, Ser. A.

[13]  Paul Camion,et al.  Antisymmetric difference sets , 1972 .

[14]  Wojciech Peisert.  All Self-Complementary Symmetric Graphs , 2001 .

[15]  Siu Lun Ma,et al.  Partial Difference Sets with Paley Parameters , 1995 .

[16]  Cunsheng Ding,et al.  A family of skew Hadamard difference sets , 2006, J. Comb. Theory, Ser. A.

[17]  Siu Lun Ma.  Partial difference sets , 1984, Discret. Math..