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 $\mathcal{NWRF}$ (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.

翻译：具有少重量的线性码在秘密共享方案、认证码、结合方案和强正则图中具有重要应用。构造线性码的方法众多，其中之一基于函数实现。此外，从函数出发的两种通用构造方法（称为第一类通用构造和第二类通用构造）引起了学者们的研究兴趣。近期，在文献\cite{Nian}中，Li和Mesnager提出了两个开放性问题：分别基于第一类和第二类通用构造，利用非弱正则坪函数构造线性码并确定其重量分布。受这两个开放性问题的启发，本文首先基于第一类通用构造，利用非弱正则坪函数构造了一些三重量和至多五重量的线性码，并确定了所构造码的重量分布。接着，基于第二类通用构造，利用属于$\mathcal{NWRF}$（本文定义）的非弱正则坪函数构造了一些三重量和至多五重量的线性码，并确定了所构造码的重量分布。我们还给出了基于第二类通用构造所获码的穿刺码，并确定了其重量分布。同时，得到了一些最优和几乎最优的线性码。此外，根据Ashikhmin-Barg条件，所构造的码在几乎所有情况下都是极小的，并基于其对偶码获得了一些具有良好访问结构的秘密共享方案。