Bent functions and their connections to combinatorics (Q2875856)

This paper gives an excellent overview of some recent results on Boolean and \(p\)-ary bent functions and their connections to combinatorics.NEWLINENEWLINEAfter an introduction, definitions and discussing basic properties of bent functions like (weak) regularity, in Section 3 several known classes of bent functions are discussed, including the Maiorana-McFarland class, Boolean partial spread bent functions, (an extension of) Dillon`s class \(H\) which in univariate representation are called Niho bent functions. The one-to-one correspondence between bent functions in the latter class and o-polynomials respectively hyperovals in \(\mathrm{PG}(2,2^m\)) is pointed out. In the last part of Section 3, the known univariate (nonquadratic) polynomials representing infinite classes of \(p\)-ary bent functions, all of which are (weakly) regular, are listed. A sporadic example of a non-weakly regular bent function is given.NEWLINENEWLINEIn Section 4, the connections between bent functions, nonlinearity and the covering radius of the first order Reed-Muller code are recalled.NEWLINENEWLINEIn Section 5, the relations between Boolean bent functions, and Hadamard difference sets and matrices, and the relations between arbitrary bent functions and relative difference sets are discussed. A recently presented construction of partial difference sets and strongly regular graphs from weakly regular \(p\)-ary bent functions is described (for \(p=3\)).NEWLINENEWLINESection 6 is on crosscorrelation of \(m\)-sequences. Amongst others, relations with Almost Bent functions are given.NEWLINENEWLINEThe area of bent functions and related functions and their connections with objects from combinatorics is a very vivid research area. Since the publication of this article several new results appeared which are strongly related to those summarized in the article. Examples are the construction of \(p\)-ary partial spread bent functions in [\textit{P. Lisoněk} and \textit{H. Y. Lu}, Des. Codes Cryptography 73, No. 1, 209--216 (2014; Zbl 1355.94104)], of which some functions in the \(p\)-ary \(PS_{ap}\) class are explicitly given in univariate form in [\textit{Nian Li}, \textit{Xiaohu Tang} and the authors, IEEE Trans Inform. Theory 59, 1818--1831 (2013)], the construction of infinite classes of not weakly regular bent functions and their analysis in [\textit{A. Çeşmelioğlu} et al., J. Comb. Theory, Ser. A 119, No. 2, 420--429 (2012; Zbl 1258.94034), Adv. Math. Commun. 7, No. 4, 425--440 (2013; Zbl 1329.94056)] and references therein, or the introduction and analysis of a concept of PN-functions for characteristic 2 in [\textit{Y. Zhou}, J. Comb. Des. 21, No. 12, 563--584 (2013; Zbl 1290.05038)] and [\textit{K.-U. Schmidt} and \textit{Y. Zhou}, J. Algebr. Comb. 40, No. 2, 503--526 (2014; Zbl 1319.51008)].NEWLINENEWLINEFor the entire collection see [Zbl 1286.05002].