Wun-Seng Chou

Value sets of Dickson polynomials over finite fields

Abstract Let Fq denote the finite field of order q where q is a prime power. If a ∈ Fq and d ≥ 1 is an integer, define the Dickson polynomial g d (x, a) = ∑ t=0 [ d 2 ] ( d (d−t) )( t d−t )(−a t x d−2t . Let {gd(x, a) | x ∈ Fq} denote the image or value set of the polynomial gd(x, a). In this paper we determine the cardinality of the value set for the Dickson polynomial gd(x, a) over the finite field Fq.

On inversive maximal period polynomials over finite fields

Over finite field GF(q) withq a power of an odd primep, we characterize inversive maximal period polynomials in terms of polynomials of orderq + 1, and then we study some properties of polynomials of orderq + 1.

Generating linear spans over finite fields

(1) sk = { vk if k ≤ n , ∑n−1 i=0 aisk−n+i if k > n , consists of all nonzero elements of V for k = 1, . . . , p − 1. Such generating patterns are of interest because they provide simple algorithms for generating the linear span of independent subsets of vector spaces over Fp (see [1] for details). In this paper we generalize a number of the results from [1] by working over Fq where Fq is the finite field of order q and by showing that if a0 6= 0, (a0, . . . , an−1) is an n-dimensional generating pattern over Fq if and only if f(x) = x − ∑n−1 i=0 aix i is a primitive polynomial over Fq. More generally, we show that the number of distinct elements generated by a linear recurring sequence is related to the order of its characteristic polynomial. For q = p < 10 with p ≤ 97, we indicate when one can find an optimal n-dimensional generating pattern over Fp with weight two, i.e. with two nonzero ai’s (in [1] the length is defined to be the number of nonzero ai’s but a more natural term is Hamming weight). If V is an n-dimensional vector space over Fq then V is isomorphic to Fqn as a vector space over Fq. Consequently, instead of considering vectors in V as in [1], we may assume that the elements of the sequence are in Fqn . We will make this identification throughout the remainder of the paper.

On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form a 1 x 1 m 1 + a 2 x 2 m 2 + ź + a n x n m n = c . We also show that if the value distribution of character sums ź x ź F q ź ( a x m + b x ) , a , b ź F q , is known, then one can obtain the number of solutions of the system of equations { x 1 + x 2 + ź + x n = α x 1 m + x 2 m + ź + x n m = β , for some particular m. We finally apply our results to induce some facts about Warings problems and the covering radius of certain cyclic codes.

On a Class of Combinatorial Sums Involving Generalized Factorials

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

On the lattice test for inversive congruential pseudorandom numbers

An important tool for the analysis of inversive congruential pseudorandom numbers is the lattice test. For a given prime modulus p, the optimal behavior of inversive congruential generators occurs when they pass the (p - 2)-dimensional lattice test. We use the connection with permutation polynomials to establish several criteria for passing the (p - 2)-dimensional lattice test. We also prove that if p is a Mersenne prime, then there exists an inversive congruential generator which has period length p and passes the (p - 2)-dimensional lattice test.

An extension of Fleck-type Möbius function and inversion

A constructive generalization of Fleck-type Mobius function and inversion via convolution polynomials is presented here. Also provided are certain related propositions and some particular examples including a few classical results as consequences.