Pencils of Quadratic Forms over Finite Fields

A formula for the number of common zeros of a non-degenerate pencil of quadratic forms is given. This is applied to pencils which count binary strings with an even number of 1\u27s prescribed distances apart

Irreducible Polynomials over GF(2) with Three Prescribed Coefficients

For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of xn-1, xn-2 and xn-3 are specified in advance. Formulas for the number of elements in GF(2n) with the first three traces specified are also given

Generalized Reciprocals, Factors of Dickson Polynomials and Generalized Cyclotomic Polynomials over Finite Fields

We give new descriptions of the factors of Dickson polynomials over finite fields in terms of cyclotomic factors. To do this generalized reciprocal polynomials are introduced and characterized. We also study the factorization of generalized cyclotomic polynomials and their relationship to the factorization of Dickson polynomials

Sums of Gauss Sums and Weights of Irreducible Codes

We develop a matrix approach to compute a certain sum of Gauss sums which arises in the study of weights of irreducible codes. A lower bound on the minimum weight of certain irreducible codes is given

Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields

We give, over a finite field Fq, explicit factorizations into a product of irreducible polynomials, of the cyclotomic polynomials of order 3·2n, the Dickson polynomials of the first kind of order 3·2n and the Dickson polynomials of the second kind of order 3·2n  − 1

Dickson polynomials over finite fields

AbstractIn this paper we introduce the notion of Dickson polynomials of the (k+1)-th kind over finite fields Fpm and study basic properties of this family of polynomials. In particular, we study the factorization and the permutation behavior of Dickson polynomials of the third kind

Factors of Dickson Polynomials over Finite Fields

We give new descriptions of the factors of Dickson polynomials over finite fields

Reversed Dickson polynomials over finite fields

AbstractReversed Dickson polynomials over finite fields are obtained from Dickson polynomials Dn(x,a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on Fq with q<200, with only one exception, all belong to the RDPP families established in this paper

Double Arrays, Triple Arrays and Balanced Grids

Triple arrays are a class of designs introduced by Agrawal in 1966 for two-way elimination of heterogeneity in experiments. In this paper we investigate their existence and their connection to other classes of designs, including balanced incomplete block designs and balanced grids