Show/Hide Menu
Hide/Show Apps
Logout
Türkçe
Türkçe
Search
Search
Login
Login
OpenMETU
OpenMETU
About
About
Open Science Policy
Open Science Policy
Communities & Collections
Communities & Collections
Help
Help
Frequently Asked Questions
Frequently Asked Questions
Guides
Guides
Thesis submission
Thesis submission
MS without thesis term project submission
MS without thesis term project submission
Publication submission with DOI
Publication submission with DOI
Publication submission
Publication submission
Supporting Information
Supporting Information
General Information
General Information
Copyright, Embargo and License
Copyright, Embargo and License
Contact us
Contact us
Studies on non-weakly regular bent functions and related structures
Date
2020
Author
Pelen, Rumi Melih
Metadata
Show full item record
This work is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
.
Item Usage Stats
323
views
0
downloads
Cite This
Interest in bent functions over finite fields arises both from mathematical theory and practical applications. There has been lots of literature addressing various properties of bent functions. They have a number of applications consisting of coding theory, cryptography, and sequence designs. They’re divided into four subclasses: regular bent functions that are contained within the class of weakly regular bent functions that are contained within the class of dual-bent functions. Additionally, there are non-weakly regular bent functions with no intersection with weakly regular, but an intersection with the class of dual-bent functions. The present thesis studies various combinatorial properties of non-weakly regular bent functions over finite fields. The principal result in the thesis is the solution of the open problem "Is there any non-weakly regular bent function f for which the dual f^* is weakly regular?" which is proposed by Çeşsmelioğlu, Meidl and Pott. We also generalize this result to plateaued functions. For an arbitray non-weakly regular bent function f, we define the partition B_+(f) and B_-(f) of F_p^n. Then, we show that, if the corresponding partition for a non-weakly regular bent function in the GMMF class gives a partial difference set then it is trivial. Moreover, we exhibit that these subsets associated with the two of the recognized sporadic examples of non-weakly regular bent functions correspond to non-trivial partial difference sets, therefore, correspond to non-trivial strongly regular graphs. For the ternary non-weakly regular bent functions in a subclass of the GMMF class, we also represent a construction method of two infinite families of translation association schemes of classes 5 and 6 in odd and even dimensions respectively. Furthermore, fusing the first or last three non-trivial relations of those association schemes we obtain association schemes of classes 3 and 4. Finally, for a non-weakly regular bent function f satisfying certain conditions, we construct three-weight linear codes on the subsets B_+(f) and B_-(f) by using one of the known conventional construction methods. Moreover, we determine the weight distribution of the corresponding three-weight linear codes in the case of f belongs to a subclass of the GMMF class. In addition to these, we prove that our construction yields minimal linear codes nearly in all cases.
Subject Keywords
Functions
,
bent
,
non-weakly regular bent
,
partial difference set
,
strongly regular graph
,
association scheme
,
linear codes
,
minimal linear codes.
URI
http://etd.lib.metu.edu.tr/upload/12625556/index.pdf
https://hdl.handle.net/11511/45790
Collections
Graduate School of Natural and Applied Sciences, Thesis
Suggestions
OpenMETU
Core
Characterizations of Partially Bent and Plateaued Functions over Finite Fields
Mesnager, Sihem; Özbudak, Ferruh; SINAK, AHMET (2018-12-30)
Partially bent and plateaued functions over finite fields have significant applications in cryptography, sequence theory, coding theory, design theory and combinatorics. They have been extensively studied due to their various desirable cryptographic properties. In this paper, we study on characterizations of partially bent and plateaued functions over finite fields, with the aim of clarifying their structure. We first redefine the notion of partially bent functions over any finite field Fq , with q a prim...
Quasilinear differential equations with strongly unpredictable solutions
Akhmet, Marat; Zhamanshin, Akylbek (2020-01-01)
The authors discuss the existence and uniqueness of asymptotically stable unpredictable solutions for some quasilinear differential equations. Two principal novelties are in the basis of this research. The first one is that all coordinates of the solution are unpredictable functions. That is, solutions are strongly unpredictable. Secondly, perturbations are strongly unpredictable functions in the time variable. Examples with numerical simulations are presented to illustrate the theoretical results.
Unpredictable solutions of linear differential and discrete equations
Akhmet, Marat; Tleubergenova, Madina; Zhamanshin, Akylbek (2019-01-01)
The existence and uniqueness of unpredictable solutions in the dynamics of nonhomogeneous linear systems of differential and discrete equations are investigated. The hyperbolic cases are under discussion. The presence of unpredictable solutions confirms the existence of Poincare chaos. Simulations illustrating the chaos are provided.
Efficient Three-Layer Iterative Solutions of Electromagnetic Problems Using the Multilevel Fast Multipole Algorithm
Onol, Can; Ucuncu, Arif; Ergül, Özgür Salih (2017-05-19)
We present a three-layer iterative algorithm for fast and efficient solutions of electromagnetic problems formulated with surface integral equations. The strategy is based on nested iterative solutions employing the multilevel fast multipole algorithm and its approximate forms. We show that the three-layer mechanism significantly reduces solution times, while it requires no additional memory as opposed to algebraic preconditioners. Numerical examples involving three-dimensional scattering problems are prese...
On multiplication in finite fields
Cenk, Murat; Özbudak, Ferruh (2010-04-01)
We present a method for multiplication in finite fields which gives multiplication algorithms with improved or best known bilinear complexities for certain finite fields. Our method generalizes some earlier methods and combines them with the recently introduced complexity notion (M) over cap (q)(l), which denotes the minimum number of multiplications needed in F-q in order to obtain the coefficients of the product of two arbitrary l-term polynomials modulo x(l) in F-q[x]. We study our method for the finite ...
Citation Formats
IEEE
ACM
APA
CHICAGO
MLA
BibTeX
R. M. Pelen, “Studies on non-weakly regular bent functions and related structures,” Thesis (Ph.D.) -- Graduate School of Natural and Applied Sciences. Mathematics., Middle East Technical University, 2020.