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Studies on non-weakly regular bent functions and related structures
Date
2020
Author
Pelen, Rumi Melih
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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
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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.