Cyclotomic difference sets in finite fields

Abstract

The classical problem of whether mth-powers with or without zero in a finite field F q form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this paper new necessary and sufficient conditions are established including those via a system of polynomial equations on Gauss sums. The author thereby solves the problem for even q which is neglected in the literature, and extends the nonexistence list for even m up to 22. Moreover, conjectures toward the complete classification are posed.

9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

“…We mention that in a recent paper [104], Xia posed essentially the same conjectures as the above folklore conjectures. (It seems that Xia was unaware of the existence of the folklore conjectures above.)…”

“…When N is large, Lehmer's theorem is not very useful since the cyclotomic numbers involved are difficult to compute; instead Gauss sums and Jacobi sums have proved to be more effective. In a recent paper [104], by using Jacobi sums and extensive Gröbner basis computations of certain overdetemined polynomial systems, Xia proved the following theorem. is a difference set in (F q , +) have also been motivated by questions in finite geometry.…”

“…In Section 2, we survey both classical and recent results on difference sets related to cyclotomy. The highlights are some recent results of Xia [104] on the long-standing conjecture that if C (N,q) 0 is a difference set in (F q , +), then N is a power of 2; and the constructions of skew Hadamard difference sets by Feng and the third author [48] by using unions of cyclotomic classes. In Section 3, we give a brief survey of results on sequences with low correlation which are related to cyclotomy.…”

On Flag‐Transitive Symmetric Designs of Affine Type

“…In particular, when C 0 is a difference set, C 0 is called ith cyclotomic !difference set and has parameters . More information on cyclotomic difference sets, including some recent results on them, are contained in .…”

“…Recently, Xia , Theorem 5.1, has proven that there are no i th cyclotomic difference sets for and , or 8.…”