9 research outputs found
Linear codes with few weights from non-weakly regular plateaued functions
Linear codes with few weights have significant applications in secret sharing
schemes, authentication codes, association schemes, and strongly regular
graphs. There are a number of methods to construct linear codes, one of which
is based on functions. Furthermore, two generic constructions of linear codes
from functions called the first and the second generic constructions, have
aroused the research interest of scholars. Recently, in \cite{Nian}, Li and
Mesnager proposed two open problems: Based on the first and the second generic
constructions, respectively, construct linear codes from non-weakly regular
plateaued functions and determine their weight distributions.
Motivated by these two open problems, in this paper, firstly, based on the
first generic construction, we construct some three-weight and at most
five-weight linear codes from non-weakly regular plateaued functions and
determine the weight distributions of the constructed codes. Next, based on the
second generic construction, we construct some three-weight and at most
five-weight linear codes from non-weakly regular plateaued functions belonging
to (defined in this paper) and determine the weight
distributions of the constructed codes. We also give the punctured codes of
these codes obtained based on the second generic construction and determine
their weight distributions. Meanwhile, we obtain some optimal and almost
optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that
the constructed codes are minimal for almost all cases and obtain some secret
sharing schemes with nice access structures based on their dual codes.Comment: 52 pages, 34 table
Value Distributions of Perfect Nonlinear Functions
In this paper, we study the value distributions of perfect nonlinear
functions, i.e., we investigate the sizes of image and preimage sets. Using
purely combinatorial tools, we develop a framework that deals with perfect
nonlinear functions in the most general setting, generalizing several results
that were achieved under specific constraints. For the particularly interesting
elementary abelian case, we derive several new strong conditions and
classification results on the value distributions. Moreover, we show that most
of the classical constructions of perfect nonlinear functions have very
specific value distributions, in the sense that they are almost balanced.
Consequently, we completely determine the possible value distributions of
vectorial Boolean bent functions with output dimension at most 4. Finally,
using the discrete Fourier transform, we show that in some cases value
distributions can be used to determine whether a given function is perfect
nonlinear, or to decide whether given perfect nonlinear functions are
equivalent.Comment: 28 pages. minor revisions of the previous version. The paper is now
identical to the published version, outside of formattin
A Further Study of Vectorial Dual-Bent Functions
Vectorial dual-bent functions have recently attracted some researchers'
interest as they play a significant role in constructing partial difference
sets, association schemes, bent partitions and linear codes. In this paper, we
further study vectorial dual-bent functions , where , denotes an
-dimensional vector space over the prime field . We give new
characterizations of certain vectorial dual-bent functions (called vectorial
dual-bent functions with Condition A) in terms of amorphic association schemes,
linear codes and generalized Hadamard matrices, respectively. When , we
characterize vectorial dual-bent functions with Condition A in terms of bent
partitions. Furthermore, we characterize certain bent partitions in terms of
amorphic association schemes, linear codes and generalized Hadamard matrices,
respectively. For general vectorial dual-bent functions with and , we give a necessary and sufficient condition on constructing
association schemes. Based on such a result, more association schemes are
constructed from vectorial dual-bent functions
Duals of non-weakly regular bent functions are not weakly regular and generalization to plateaued functions
It is known that the dual of a weakly regular bent function is again weakly regular. On the other hand, the dual of a nonweakly regular bent function may not even be a bent function. In 2013, cesmelioglu, Meidl and Pott pointed out that the existence of a non-weakly regular bent function having weakly regular bent dual is an open problem. In this paper, we prove that for an odd prime p and n is an element of Z+, if f : F-p(n)-> F-p is a non-weakly regular bent function such that its dual f* is bent, then f**(-x) = f(x), and f* is non-weakly regular, which solves the open problem. We also generalize our results to plateaued functions
A Further Study of Vectorial Dual-Bent Functions
Vectorial dual-bent functions have recently attracted some researchers\u27 interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions and linear codes. In this paper, we further study vectorial dual-bent functions , where , denotes an -dimensional vector space over the prime field . We give new characterizations of certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A) in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. When , we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Furthermore, we characterize certain bent partitions in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. For general vectorial dual-bent functions with and , we give a necessary and sufficient condition on constructing association schemes. Based on such a result, more association schemes are constructed from vectorial dual-bent functions