由非弱正则plateaued函数构造的少重量线性码
Linear codes with few weights from non-weakly regular plateaued functions
AI总结：
本文针对Li和Mesnager提出的公开问题，基于两种通用构造法从非弱正则plateaued函数构造少重量线性码，确定其重量分布，并得到最优码、极小码及相应秘密共享方案。
摘要：
少重量线性码在秘密共享方案、认证码、结合方案和强正则图等领域具有重要应用。构造线性码的方法有很多，其中之一是基于函数的方法。此外，基于函数构造线性码的两种通用构造方法，即第一通用构造法和第二通用构造法，引起了学者们的研究兴趣。最近，在文献\cite{Nian}中，Li和Mesnager提出了两个公开问题：分别基于第一通用构造法和第二通用构造法，从非弱正则plateaued函数构造线性码并确定其重量分布。受这两个公开问题的启发，本文首先基于第一通用构造法，从非弱正则plateaued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，基于第二通用构造法，我们从属于$\mathcal{NWRF}$（本文中定义）的非弱正则plateaued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通用构造法得到的这些码的缩短码，并确定了它们的重量分布。同时，我们得到了一些最优和几乎最优的线性码。此外，根据Ashikhmin-Barg条件，我们证明了所构造的码在几乎所有情形下都是极小码，并基于它们的对偶码得到了一些具有良好访问结构的秘密共享方案。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
