Linear codes with few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. \cite{DD2015} and Zhou et al. \cite{ZLFH2015} to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves the open problem of Ding et al. \cite{DD2015}. Further, this paper constructs new linear codes with two or three weights and presents the weight distributions of these codes. They contains some optimal codes meeting certain bound on linear codes.
There are infinitely many bent functions for which the dual is not bent
Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs since their duals are also bent functions. In general this does not apply to non-weaky regular bent functions. However, the first known construction of non-weakly regular bent functions by Ceşmelioğlu et { \it al.}, 2012, yields bent functions for which the dual is also bent. In this paper the first construction of non-weakly regular bent functions for which the dual is not bent is pr…
In this paper, we study the dual of generalized bent functions $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$ -dimensional vector space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The dual of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of \cite{Cesmelioglu5} , we obtain an e…
Strongly regular graphs from weakly regular plateaued functions
本文首次给出了奇特征有限域上弱正则平台函数的强正则图和结合方案的构造。我们将Chee等人在[Journal of Algebras Combinatorics，34（2），251-266，2011]中给出的弱正则bent函数的强正则图的构造方法推广到弱正则plateaued函数。在此框架下，我们从弱正则平台函数构造了具有三类参数的强正则图，并给出了它们的齐次条件。我们还从弱正则p元平台函数构造了一族p类结合方案。
Some New Constructions of Generalized Plateaued Functions
Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In \cite{Mesnager9} , Mesnager \emph{et al.} introduced generalized plateaued functions in order to study plateaued functions in the general context of generalized $p$ -ary functions. In this paper, we focus on the constructions of generalized $p$ -ary $s$ -plateaued functions from $V_{n}$ to $\mathbb{Z}_{p^k}$ , where $V_{n}$ is an $n$ -dimensional vector space over $\mathbb{F}_{p}$ , $p$ is a prime, $k\geq 1$ and $n+s$ is even when $p=2$ . In particular, …
A new class of three-weight linear codes from weakly regular plateaued functions
低权线性码在秘密共享、认证码、通信和强正则图等领域有着广泛的应用。本文考虑任意特征的三重线性码。为此，我们推广了Mesnager在[Cryptography and Communications 9（1），71-84，2017]中给出的最新贡献。首先，我们提出了一类新的二元线性码的三个重量从坪布尔函数和重量分布。接下来，我们介绍了奇特征$p$中的（弱）正则平台函数的概念，并给出了这些函数的具体例子。此外，我们构造了一类新的三重线性$p$ -ary码，并确定了它们的重量分布。最后分析了所构造的秘密共享方案的线性码。
