Studies on non-weakly regular bent functions and related structures

Pelen, Rumi Melih
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.


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 the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems
AKBAŞ, MERAL; Kaya, Serap; Kaya Merdan, Songül (2017-07-01)
We prove long-time stability of linearly extrapolated BDF2 (BDF2LE) timestepping methods, together with finite element spatial discretizations, for incompressible Navier-Stokes equations (NSE) and related multiphysics problems. For the NSE, Boussinesq, and magnetohydrodynamics schemes, we prove unconditional long time L-2 stability, provided external forces (and sources) are uniformly bounded in time. We also provide numerical experiments to compare stability of BDF2LE to linearly extrapolated Crank-Nicolso...
New bent functions from permutations and linear translators
MESNAGER, sihem; ONGAN, pınar; Özbudak, Ferruh (2017-04-12)
Starting from the secondary construction originally introduced by Carlet ["On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities", Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 2006], that we shall call "Carlet` ssecondary construction", Mesnager has showed how one can construct several new primary constructions of bent functions. In particular, she has showed that three tuples of permutations over the finite field F2m such that the inverse of their sum...
Citation Formats
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.