Classification of some quadrinomials over finite fields of odd characteristic

Date

2023-03-01

Author

Özbudak, Ferruh
Gulmez Temur, B. or Temur

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

214
views

0
downloads

In this paper, we completely determine all necessary and sufficient conditions such that the polynomialf(x)=x3+axq+2+bx2q+1+cx3q" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�(�)=�3+��+2+��2�+1+��3�, wherea,b,c&#x2208;Fq&#x204E;" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">⁎�,�,�∈��⁎, is a permutation quadrinomial ofFq2" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">��2over any finite field of odd characteristic. This quadrinomial has been studied first in[25]by Tu, Zeng and Helleseth, later in[24]Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, wherechar(Fq)=2" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�ℎ��(��)=2and finally, in[16], Li, Qu, Li and Chen proved that the sufficient condition given in[24]is also necessary and thus completed the solution in even characteristic case. In[6]Gupta studied the permutation properties of the polynomialx3+axq+2+bx2q+1+cx3q" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�3+��+2+��2�+1+��3�, wherechar(Fq)=3,5" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�ℎ��(��)=3,5anda,b,c&#x2208;Fq&#x204E;" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">⁎�,�,�∈��⁎and proposed some new classes of permutation quadrinomials ofFq2" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">��2.In particular, in this paper we classify all permutation polynomials ofFq2" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">��2of the formf(x)=x3+axq+2+bx2q+1+cx3q" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�(�)=�3+��+2+��2�+1+��3�, wherea,b,c&#x2208;Fq&#x204E;" role="presentation" style="margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">⁎�,�,�∈��⁎, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.

URI

https://hdl.handle.net/11511/102390

Journal

FINITE FIELDS AND THEIR APPLICATIONS

DOI

https://doi.org/10.1016/j.ffa.2022.102158

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

Classification of skew-symmetric forms corresponding to cluster algebras with principal coefficients Mazı, Sedanur; Seven, Ahmet İrfan; Department of Mathematics (2016) In this thesis, we study algebraic and combinatorial properties of the skew-symmetric forms that correspond to cluster algebras with principal coefficients. We obtain a classification of these forms under congruence and compute the Arf invariants for finite types.
Topology of the complement of a real algebraic curve in ℂP2 Finashin, Sergey (1984-07-01) In this paper we consider the problem of the disposition of the set of points of a nonsingular real algebraic curve of given degree in ℂP2. The homotopy description of the complement of such a curve in ℂP2 is the first step toward solving the problem of disposition mentioned. In the case of an arbitrary curve we are able to prove that the complement indicated is homotopy equivalent with a three-dimensional cell complex of special form. For a certain class of curves the complex turns out to be two-dimensiona...
Stability of differential equations with piecewise constant arguments of generalized type Akhmet, Marat (Elsevier BV, 2008-02-15) In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367-383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and Sufficient conditions for stability of the zero Solution are ob...
Multiplication modules which are distributive Erdoğdu, Vahap (Elsevier BV, 1988-10) We prove results which include necessary and sufficient conditions for a multiplication module to be distributive
Characterisation and enumeration of a class of semi bent quadratic Boolean functions KOÇAK, Neşe; Koçak, Onur Ozan; Özbudak, Ferruh; SAYGI, ZÜLFÜKAR (2015-01-01) In this paper, we consider semi-bentness of quadratic Boolean functions defined for even n and give the characterisation of these functions. Up to our knowledge, semi-bentness of this class has not been investigated before and we proved that semi-bent functions of this form exist only for 6|n. Furthermore, we present a method for enumeration of semi-bent and bent functions in certain classes. Using this method we find the exact number of semi-bent functions of this form. Moreover, we complete some previous ...

Citation Formats

F. Özbudak and B. o. T. Gulmez Temur, “Classification of some quadrinomials over finite fields of odd characteristic,” FINITE FIELDS AND THEIR APPLICATIONS, vol. 87, pp. 102158–102158, 2023, Accessed: 00, 2023. [Online]. Available: https://hdl.handle.net/11511/102390.