| In this dissertation we consider two different but related objects: skew Hadamard difference sets and p-ary bent functions. A difference set D in a group G is called a skew Hadamard difference set (in short, SHDS) if G is the disjoint union of D, the set of inverses of D, and the identity element. We construct a new family of skew Hadamard difference sets in the additive group of F32h+1 , by using a class of permutation polynomials of F32h+1 obtained from the Ree-Tits slice symplectic spreads in the three-dimensional projective space over F32h+1 . Also we generalize the recent Ding-Yuan construction of SHDS by using planar functions. This generalization also gives rise to new constructions of strongly regular graphs. All SHDS that are considered in this dissertation live in elementary abelian groups, more specifically, in the additive group of presemifields.;In our second project we study the so-called p-ary bent functions, a special type of function with various applications in coding theory and cryptography. We prove two conjectures on ternary weakly regular bent functions.;From this dissertation it will be clear that planar functions provide a link between the two main topics of this work. In both projects, our major tools are Gauss sums, Stickelberger's theorem on Gauss sums and p -adic number theory. |
