In this paper, several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions are constructed based on a generic construction method. Instead of the whole space, we use the subspaces $B_+(f)$ or $B_-(f)$ associated with a ternary non-weakly regular dual-bent function $f$. Unusually, we use the pre-image sets of the dual function $f^*$ in $B_+(f)$ or $B_-(f)$ as the defining sets of the corresponding codes. Since the size of the defining sets of the constructed codes is flexible, it enables us to construct several codes with different parameters for a fixed dimension. We represent the weight distribution of the constructed codes. We also give several examples.
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, …
