Related papers: Further results on some classes of permutation pol…

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation…

By the Fourier transformations, any group-invariant functions over finite Abelian groups are transformed into group-invariant functions over the character groups. In this paper, we calculate matrix elements of this transformations under…

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$ be the polynomial defined by the functional equation $\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$. When is $g_{n,q}$ a permutation…

In this paper we study the distribution of the size of the value set for a random polynomial with degree at most $q-1$ over a finite field $\mathbb{F}_q$. We obtain the exact probability distribution and show that the number of missing…

Let $q$ be a prime power and $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $n>1$. We introduce the class of the linearized polynomials $L(x)$ over $\mathbb F_{q^n}$ such that…

We give new characterizations of the algebra $\mathscr{L}_n(\mathbb{F}_{q^n})$ formed by all linearized polynomials over the finite field $\mathbb{F}_{q^n}$ after briefly surveying some known ones. One isomorphism we construct is between…

In this paper, we consider the new q-extension of Frobenius-Euler numbers and polynomials and we derive some interesting identities from the orthogonality type properties for the new q-extension of Frobenius-Euler polynomials. Finally we…

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a…

We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study…

This research is aimed to give a determinantal definition for the $q$-Appell polynomials and show some classical properties as well as find some interesting properties of the mentioned polynomials in the light of the new definition.

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs…

In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over $\mathbb{F}_{q^{2}}$. The paper…

In this paper, we investigate the properties of q-Hermite polynomials related to q-Bernstein polynomials. From these properties, we derive some interesting relations between q-Berstein polynomials and q-Hermite polynomials.

We estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the…

In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in…

Let $f={\tt X}^r(a+{\tt X}^{2(q-1)})\in{\Bbb F}_{q^2}[{\tt X}]$, where $a\in{\Bbb F}_{q^2}^*$ and $r\ge 1$. The parameters $(q,r,a)$ for which $f$ is a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ have been determined in the following…

Let $\mathbb{F}$ be a finite field and let $f$ be a linear polynomial in $\mathbb{F}[x]$. We investigate the number of polynomials of degree $d$ which commute with $f$ under composition. In so doing, we rediscover a result of Park, but with…

Let $q>2$, and let $a$ and $b$ be two elements of the finite field $\mathbb{F}_q$ with $a\ne 0$. Carlitz represented the transposition $(0a)$ by a polynomial of degree $(q-2)^3$. In this note, we represent the transposition $(ab)$ by a…

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod…

The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic…