Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many construction methods for linear codes from functions have been proposed in the literature. In this paper, we generalize the recent construction methods given by Tang et al. in [IEEE Transactions on Information Theory, 62(3), 1166-1176, 2016] to weakly regular plateaued functions over finite fields of odd characteristic. We first construct three-weight linear codes from weakly regular plateaued functions based on the second generic construction and then determine their weight distributions. We also give a punctured version and subcode of each constructed code. We note that they may be (almost) optimal codes and can be directly employed to obtain (democratic) secret sharing schemes, which have diverse applications in the industry. We next observe that the constructed codes are minimal for almost all cases and finally describe the access structures of the secret sharing schemes based on their dual codes.

中文翻译：

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弱正则平稳函数的几类极小权重线性码

最小线性码在秘密共享方案和安全两方计算中具有重要应用。有几种构造线性代码的方法，其中一种方法是基于有限域上的函数。最近，文献中提出了许多基于函数的线性码构造方法。在本文中，我们概括了 Tang 等人最近给出的构造方法。在 [IEEE Transactions on Information Theory, 62(3), 1166-1176, 2016] 到奇特性的有限域上的弱正则平稳函数。我们首先基于第二个通用构造从弱正则平稳函数构造三权重线性代码，然后确定它们的权重分布。我们还给出了每个构造代码的穿孔版本和子代码。我们注意到它们可能是（几乎）最优代码，可以直接用于获得（民主）秘密共享方案，在行业中有多种应用。我们接下来观察到几乎所有情况下构造的代码都是最小的，最后描述了基于双重代码的秘密共享方案的访问结构。