Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDegree of Orthomorphism Polynomials over Finite Fields
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An orthomorphism over a finite field $mathbb{F}_q$ is a permutation $theta:mathbb{F}_qmapstomathbb{F}_q$ such that the map $xmapstotheta(x)-x$ is also a permutation of $mathbb{F}_q$. The degree of an orthomorphism of $mathbb{F}_q$, that is, the degree of the associated reduced permutation polynomial, is known to be at most $q-3$. We show that this upper bound is achieved for all prime powers $q otin{2, 3, 5, 8}$. We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct $3$-homogeneous Latin bitrades.
rate research
Read More
In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.
We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called {it nonsingular} if $gcd(|G|, a) = 1$ for any $ain M$. In this paper, we focus our study on nonsingular splittings of cyclic groups. We introduce a new notation --direct KM logarithm and we prove that if there is a prime $q$ such that $M$ splits $mathbb{Z}_q$, then there are infinitely many primes $p$ such that $M$ splits $mathbb{Z}_p$.
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field.
In 2005, Kayal suggested that Schoofs algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pilas algorithm for abelian varieties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
