19 research outputs found
On the dual of the dual hyperoval from APN function f(x)=x3+Tr(x9)
AbstractUsing a quadratic APN function f on GF(2d+1), Yoshiara (2009) [15] constructed a d-dimensional dual hyperoval Sf in PG(2d+1,2). In Taniguchi and Yoshiara (2005) [13], we prove that the dual of Sf, which we denote by Sfâ„, is also a d-dimensional dual hyperoval if and only if d is even. In this note, for a quadratic APN function f(x)=x3+Tr(x9) on GF(2d+1) by Budaghyan, Carlet and Leander (2009) [2], we show that the dual Sfâ„ and the transpose of the dual Sfâ„T are not isomorphic to the known bilinear dual hyperovals if d is even and dâ©Ÿ6
A new construction of the d-dimensional BurattiâDel Fra dual hyperoval
AbstractThe BurattiâDel Fra dual hyperoval Dd(F2) is one of the four known infinite families of simply connected d-dimensional dual hyperovals over F2 with ambient space of vector dimension (d+1)(d+2)/2 (Buratti and Del Fra (2003) [1]). A criterion (Proposition 1) is given for a d-dimensional dual hyperoval over F2 to be covered by Dd(F2) in terms of the addition formula. Using it, we provide a simpler model of Dd(F2) (Proposition 3). We also give conditions (Lemma 4) for a collection S[B] of (d+1)-dimensional subspaces of KâK constructed from a symmetric bilinear form B on Kâ
F2d+1 to be a quotient of Dd(F2). For when d is even, an explicit form B satisfying these conditions is given. We also provide a proof for the fact that the affine expansion of Dd(F2) is covered by the halved hypercube (Proposition 10)
A Generalization of APN Functions for Odd Characteristic
Almost perfect nonlinear (APN) functions on finite fields of characteristic
two have been studied by many researchers. Such functions have useful
properties and applications in cryptography, finite geometries and so on.
However APN functions on finite fields of odd characteristic do not satisfy
desired properties. In this paper, we modify the definition of APN function in
the case of odd characteristic, and study its properties
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented