On homogeneous planar functions

Let $p$ be an odd prime and \F_q be the finite field with $q=p^n$ elements. A planar function f:\F_q\rightarrow\F_q is called homogenous if $f(\lambda x)=\lambda^df(x)$ for all \lambda\in\F_p and x\in\F_q, where $d$ is some fixed positive integer. We characterize $x^2$ as the unique homogenous planar function over \F_{p^2} up to equivalence.Comment: Introduction modified to: 1. give the correct definition of equivalence, 2. add some references. Other part unaltere

Cyclotomic Constructions of Skew Hadamard Difference Sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A

Strongly Regular Graphs From Unions of Cyclotomic Classes

We give two constructions of strongly regular Cayley graphs on finite fields \F_q by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.Comment: 17 pages; to appear in J. Combin. Theory (B