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Register a new userFurther results on some classes of permutation polynomials over finite fields
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Added by
Xiaogang Liu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Xiaogang Liu
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Let $mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with their studies, and get some further results about the permutation properties of the permutation polynomials. Also, some new classes of permutation polynomials are constructed. For these, we alter the coefficients, exponents or the underlying fields, etc.
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Permutation polynomials have many applications in finite fields theory, coding theory, cryptography, combinatorial design, communication theory, and so on. Permutation binomials of the form $x^{r}(x^{q-1}+a)$ over $mathbb{F}_{q^2}$ have been studied before, K. Li, L. Qu and X. Chen proved that they are permutation polynomials if and only if $r=1$ and $a^{q+1} ot=1$. In this paper, we consider the same binomial, but over finite fields $mathbb{F}_{q^3}$ and $mathbb{F}_{q^e}$. Two different kinds of methods are employed, and some partial results are obtained for them.
For the finite field $mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+delta)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the structures and properties of the field elements are analyzed.
Let $mathbb{F}_{p^{n}}$ be the finite field with $p^n$ elements and $operatorname{Tr}(cdot)$ be the trace function from $mathbb{F}_{p^{n}}$ to $mathbb{F}_{p}$, where $p$ is a prime and $n$ is an integer. Inspired by the works of Mesnager (IEEE Trans. Inf. Theory 60(7): 4397-4407, 2014) and Tang et al. (IEEE Trans. Inf. Theory 63(10): 6149-6157, 2017), we study a class of bent functions of the form $f(x)=g(x)+F(operatorname{Tr}(u_1x),operatorname{Tr}(u_2x),cdots,operatorname{Tr}(u_{tau}x))$, where $g(x)$ is a function from $mathbb{F}_{p^{n}}$ to $mathbb{F}_{p}$, $taugeq2$ is an integer, $F(x_1,cdots,x_n)$ is a reduced polynomial in $mathbb{F}_{p}[x_1,cdots,x_n]$ and $u_iin mathbb{F}^{*}_{p^n}$ for $1leq i leq tau$. As a consequence, we obtain a generic result on the Walsh transform of $f(x)$ and characterize the bentness of $f(x)$ when $g(x)$ is bent for $p=2$ and $p>2$ respectively. Our results generalize some earlier works. In addition, we study the construction of bent functions $f(x)$ when $g(x)$ is not bent for the first time and present a class of bent functions from non-bent Gold functions.
Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, etc. In this paper, we construct several new classes of permutation polynomials, and the necessities of some permutation polynomials are studied.
In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $mathbb{F}_3$ and $mathbb{F}_4$. We study the bound of LCD codes over $mathbb{F}_q$.
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