For more information about this meeting, contact Robert Vaughan.

Title: | Cyclotomic constructions of strongly regular Cayley graphs and difference sets |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Qing Xiang, University of Delaware |

Abstract: |

The idea of constructing dierence sets and strongly regular Cayley graphs
from cyclotomic classes of nite elds goes back to Paley. In the mid-
20th century, this idea was pursued vigorously by many researchers, such
as Baumert, Chowla, Hall, Lehmer, Van Lint, Schrijver, Storer, Whiteman,
Yamamoto, etc. However, this method for constructing dierence sets has
had only very limited success. Let q be a prime power and N|(q-1), N > 1.
It is known that a single cyclotomic class of order N of F_q can form a dif-
ference set in (F_q,+) if N = 2, 4 or 8 and q satisfies certain conditions. It
was conjectured that the converse is also true. Namely, if the nonzero N-th
powers of F_q form a difference set in (F_q,+), then N is necessarily 2, 4, or
8. This conjecture has been verified up to N = 20. There is a conjecture of a
similar nature for cyclotomic strongly regular graphs.
We will report new constructions of both strongly regular Cayley graphs
and skew Hadamard difference sets by using unions of cyclotomic classes of
very large orders. Implications on association schemes will be discussed. The
main tools we used are Gauss sums, instead of cyclotomic numbers. The talk
is based on joint work with Tao Feng. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 02 / 02 / 2012 |

Time: | 11:15am - 12:05pm |