1,154 research outputs found
Secondary constructions of vectorial -ary weakly regular bent functions
In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of
vectorial/Boolean bent functions via the so-called property was
introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the
construction of -ary weakly regular bent functions. The objective of this
paper is to further generalize these constructions, following the ideas in
\cite{Bapic, Zheng}, for secondary constructions of vectorial -ary weakly
regular bent and plateaued functions. We also present some infinite families of
such functions via the -ary Maiorana-McFarland class. Additionally, we give
another characterization of the property for the -ary case via
second-order derivatives, as it was done for the Boolean case in \cite{Zheng}
The Subfield Codes of Some Few-Weight Linear Codes
Subfield codes of linear codes over finite fields have recently received a
lot of attention, as some of these codes are optimal and have applications in
secrete sharing, authentication codes and association schemes. In this paper,
the -ary subfield codes of six different families of
linear codes are presented, respectively. The parameters and
weight distribution of the subfield codes and their punctured codes
are explicitly determined. The parameters of the duals of
these codes are also studied. Some of the resultant -ary codes
and their dual codes are optimal
and some have the best known parameters. The parameters and weight enumerators
of the first two families of linear codes are also settled,
among which the first family is an optimal two-weight linear code meeting the
Griesmer bound, and the dual codes of these two families are almost MDS codes.
As a byproduct of this paper, a family of quaternary
Hermitian self-dual code are obtained with . As an application,
several infinite families of 2-designs and 3-designs are also constructed with
three families of linear codes of this paper.Comment: arXiv admin note: text overlap with arXiv:1804.06003,
arXiv:2207.07262 by other author
Minimal -ary codes from non-covering permutations
In this article, we propose several generic methods for constructing minimal linear codes over the field . The first construction uses the method of direct sum of an arbitrary function and a bent function to induce minimal codes with parameters and minimum distance larger than . For the first time, we provide a general construction of linear codes from a subclass of non-weakly regular plateaued functions, which partially answers an open problem posed in [22]. The second construction deals with a bent function and a subspace of suitable derivatives of , i.e., functions of the form for some . We also provide a sound generalization of the recently introduced concept of non-covering permutations [45]. Some important structural properties of this class of permutations are derived in this context. The most remarkable observation is that the class of non-covering permutations contains the class of APN power permutations (characterized by having two-to-one derivatives). Finally, the last general construction combines the previous two methods (direct sum, non-covering permutations and subspaces of derivatives) together with a bent function in the Maiorana-McFarland class to construct minimal codes (even those violating the Ashikhmin-Barg bound) with a larger dimension. This last method proves to be quite flexible since it can lead to several non-equivalent codes, depending to a great extent on the choice of the underlying non-covering permutation
Minimal linear codes from characteristic functions
Minimal linear codes have interesting applications in secret sharing schemes
and secure two-party computation. This paper uses characteristic functions of
some subsets of to construct minimal linear codes. By properties
of characteristic functions, we can obtain more minimal binary linear codes
from known minimal binary linear codes, which generalizes results of Ding et
al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By
characteristic functions corresponding to some subspaces of , we
obtain many minimal linear codes, which generalizes results of [IEEE Trans.
Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf.
Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic
functions to present a characterization of minimal linear codes from the
defining set method and present a class of minimal linear codes
Constant mean curvature surfaces
In this article we survey recent developments in the theory of constant mean
curvature surfaces in homogeneous 3-manifolds, as well as some related aspects
on existence and descriptive results for -laminations and CMC foliations of
Riemannian -manifolds.Comment: 102 pages, 17 figure
- …