Classification of some quadrinomials over finite fields of odd characteristic

Date

2023-03-01

Author

Özbudak, Ferruh
Gulmez Temur, B. or Temur

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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

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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.