We give a characterization of permutation polynomials over a finite field based on their coefficients, similar to Hermite's Criterion. Then, we use this result to obtain a formula for the total number of monic permutation binomials of degree less than 4p over F4p+1, where p and 4p+1 are primes, in terms of the numbers of three special types of permutation binomials.
*****We are interested in generalizations of the cycle index of
decomposable objects: permutations,
polynomials over finite fields, 2-regular graphs, random mappings
and so on. By studying the generating functions of these objects
we can
learn how the behavior of a combination of components look like when
the length of the object goes to infinity.
I will introduce R-orthomorphisms of finite fields, and use them to give nontrivial bounds for certain character sums. I will give constructions for R-orthomorphisms by cyclotomic mappings.
*****Universal cycles are generalizations of de Bruijn cycles. Suppose we are given a family Fn of combinatorial objects of rank n with m = |Fn|. Assume that each F ∈ Fn is specified by some sequence [x1, x2, ..., xn], where xi ∈ A, for some fixed alphabet A. U = (a0, a1 , ..., am-1) is a universal cycle (Ucycle) for Fn if [ai+1, ..., ai+n], 0 ≤ i < m, runs through each element of Fn exactly once, where index addition is performed modulo m. Ucycles can be constructed for a variety of families of combinatorial structures including permutations, partitions and k-subsets of an n-set. In this talk we consider the existence of Ucycles for block designs. In particular, we show that Ucycles exist for every cyclic triple system with v ≥ 13 and arbitrary l. The proof method is constructive, therefore, similar techniques can be applied to BIBDs with k ≥ 3 and to PBDs.
A mixed radix Gray Code is a sequence of n-tuples (xn, xn-1, ..., x1) with 0 ≤ xi ≤ si for i=0, 1, ..., n. Let C(k:xn, xn-1, ..., x1) be the n-tuples such that ∑i=1n xi = k. We prove that the order of occurance of the fixed weight n-tuples in the standard mixed radix reflected Gray code ordering of all n-tuples is also a minimal change ordering of C(k:xn, xn-1, ..., x1). The minimal change is such that the successive n-tuples only differ in two positions and these positions differ in value from the predecessor by 1. We present an algorithm to calculate the successor in this minimal change ordering.