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Studies on non-weakly regular bent functions and related structures

2020
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.