11 research outputs found
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
ON DILLON\u27S CLASS H OF BENT FUNCTIONS, NIHO BENT FUNCTIONS AND O-POLYNOMIALS
One of the classes of bent Boolean functions introduced by John Dillon in his thesis
is family H. While this class corresponds to a nice original construction of bent functions in
bivariate form, Dillon could exhibit in it only functions which already belonged to the well-
known Maiorana-McFarland class. We first notice that H can be extended to a slightly larger
class that we denote by H. We observe that the bent functions constructed via Niho power
functions, which four examples are known, due to Dobbertin et al. and to Leander-Kholosha,
are the univariate form of the functions of class H. Their restrictions to the vector spaces
uF2n=2 , u 2 F?
2n, are linear. We also characterize the bent functions whose restrictions to the
uF2n=2 \u27s are affine. We answer to the open question raised by Dobbertin et al. in JCT A 2006
on whether the duals of the Niho bent functions introduced in the paper are Niho bent as well,
by explicitely calculating the dual of one of these functions. We observe that this Niho function
also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing
whether H (or H) is a subclass of the Maiorana-McFarland completed class. We then show that
the condition for a function in bivariate form to belong to class H is equivalent to the fact that
a polynomial directly related to its definition is an o-polynomial and we deduce eight new cases
of bent functions in H which are potentially new bent functions and most probably not affine
equivalent to Maiorana-McFarland functions