Related papers: Constructions of involutions over finite fields

A novel method for computation of the discrete Fourier transform over a finite field with reduced multiplicative complexity is described. If the number of multiplications is to be minimized, then the novel method for the finite field of…

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…

Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on…

For any given polynomial $f$ over the finite field $\mathbb{F}_q$ with degree at most $q-1$, we associate it with a $q\times q$ matrix $A(f)=(a_{ik})$ consisting of coefficients of its powers $(f(x))^k=\sum_{i=0}^{q-1}a_{ik} x^i$ modulo…

In 2019, Xiang Fan \cite{xfan} classified all permutation polynomials of degree $7$ over finite fields of odd characteristics. In this paper, we use this classification to determine the complete list of degree $7$ orthomorphism polynomials…

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…

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using…

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation…

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…

In this paper we present and analyse a construction of irreducible polynomials over odd prime fields via the transforms which take any polynomial $f \in \mathbf{F}_p[x]$ of positive degree $n$ to $\left(\frac{x}{k} \right)^n \cdot…

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic…

Let $q$ be a power of a prime and $\mathbb{F}_q$ be a finite field with $q$ elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form $cx-x^s + x^{qs}$ over $\mathbb{F}_{q^2}$, and…

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither…

Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$ over $\mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form…

For a wide family of formations $\mathfrak{F}$ it is proved that the $ \mathfrak{F}$-residual of a permutation finite group can be computed in a polynomial time. Moreover, if in the previous case $\mathfrak{F}$ is hereditary, then an…

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…

Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general…

The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and…

In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…