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A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where $\tilde{F}(x)=(f_1(x),\ldots,f_m(x))$ is a vectorial bent function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, and $a_i$, $1\le i\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $F(x) = x^{2^r}{\rm Tr^n_m}(\Lambda(x))$, where $\Lambda$ is a linearized permutation of $\mathbb{F}_{2^m}$. In the first part of this article it is shown that plateaued functions with $2^n-2^m$ bent components can have nonlinearity at most $2^{n-1}-2^{\lfloor\frac{n+m}{2}\rfloor}$, a bound which is attained by the example $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r

Zhou 2013 introduced modified planar functions to describe $(2^n,2^n,2^n,1)$ relative difference sets $R$ as a graph of a function on the finite field $\F_{2^n}$, and pointed out that projections of $R$ are difference sets that can be described by negabent or bent$_4$ functions, which are Boolean functions given in multivariate form. Objective of this paper is to contribute to the understanding of these component functions of modified planar functions. We first completely describe a multivariate version of modified planar functions in terms of their bent$_4$ components. In the second part we characterize the component functions of (univariate) modified planar functions in terms of appropriate generalizations of the Walsh-Hadamard transform, with respect to which they have a flat spectrum. We hereby obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function.

New families of complete caps in finite Galois spaces are obtained. For most pairs (N, p(h)) with h > 8 and N equivalent to 0 (mod 4), they turn out to be the smallest known complete caps in AG(N, p(h)). Our constructions rely on the bicovering properties of certain plane arcs contained in plane cubic curves with a cusp. (C) 2013 Wiley Periodicals, Inc.

Bu tezde asal karakteristikte tanımlanmış cebirsel eğriler konusundaki iki problemi ele aldık. İlk bölümde sonlu bir cisim olan Fq üzerinde tanımlı eğrileri çalıştık. Yeteri kadar büyük her tamsayı g için öngörülmüş sayıda r dereceli (r = 1,...,m) noktası olan cinsi g bir eğrinin varlığını gösterdik. Bu sonuç, belli bir dereceye kadar öngörülmüş katsayılı L-polinomu olan bir eğrinin varlığını göstermiştir. İkinci bölümde tek karakteristikli, cebirsel olarak kapalı K cismi üzerinde tanımlı eğrileri göz önüne aldık. Otomorfizma grubunun sayısı 3(2g2 + g)(√8g + 1 + 3)'den büyük olan düzlemsel düzgün bir eğrinin Hermitian eğrisine birasyonel olarak eşdeğer olduğunu gösterdik. In this thesis we consider two problems related to algebraic curves in prime characteristic. In the first part, we study curves defined over the finite field Fq. We prove that for each sufficiently large integer g there exists a curve of genus g with prescribed number of degree r points for r = 1,...,m. This leads to the existence of a curve whose L-polynomial has prescribed coefficients up to some degree. In the second part, we consider curves defined over algebraically closed fields K of odd characteristic. We show that a plane smooth curve which has a K-automorphism group of order larger than 3(2g2 + g)(√8g + 1 + 3) must be birationally equivalent to a Hermitian curve. 64