Vendredi 9 septembre 2016 à 10h30 en salle C48
Sihem Mesnager (Université Paris VIII et Télécom ParisTech)

Titre : Codes from bent functions over finite fields

Résumé :
Certain special types of functions over finite fields are closely related to linear or nonlinear codes. In the past decade, a lot of progress on interplays between special functions and codes has been made. In particular, APN functions, planar functions, Dickson polynomials, and q-polynomials were employed to construct linear codes with optimal or almost optimal parameters. Recently, several new approaches to constructing linear codes with special types of functions were proposed, and a lot of linear codes with excellent parameters were obtained. Bent functions are maximally nonlinear Boolean functions. They were introduced by Rothaus in the 1960's and initially studied by Dillon as early as 1974 in his Thesis. The notion of bent function has been extended in arbitrary characteristic and to a more general notion: the so-called plateaued functions (in the sens that the set of bent functions is a special family of plateaued functions). For their own sake as interesting combinatorial objects, but also for their relations to coding theory (e.g. Reed-Muller codes, Kerdock codes, etc.), combinatorics (e.g. difference sets), design theory, sequence theory, and applications in cryptography (design of stream ciphers and of S-boxes for block ciphers), bent functions have attracted a lot of research for the past four decades. It is well-known that Kerdock codes are constructed from bent functions. Very recently, some authors have highlighted that bent functions lead to the construction of interesting linear codes (in particular, linear codes with few weights). This talk is devoted to linear codes from bent functions and other plateaued functions. We shall present the state of the art as well as our recent contributions in this topic. We will present two generic constructions of linear codes involving special functions and investigate constructions of good linear codes based on the generic constructions involving bent functions over finite fields. More specifically, we shall give more details on our recent (2016) results on linear codes with few weights from weakly regular bent functions based on a generic construction.