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.