弱正则Plateaued平衡函数的极小线性码
Minimal Linear Codes From Weakly Regular Plateaued Balanced Functions
摘要：
线性码在秘密共享方案、安全两方计算、关联方案、强正则图、认证码和通信等方面有着广泛的应用。文献中有大量的小权线性码，但很少有极小权线性码。本文首次将奇特征有限域上的弱正则平台平衡函数用于线性码的第二类构造方法。本文的主要结果如下。我们首先从弱正则平衡函数出发，构造了几个带可变参数的三重和四重线性码。值得注意的是，（几乎）最佳的代码可以从这些功能。我们还观察到，本文所得到的所有码都是极小码，因此它们可以直接用来构造具有高度民主性的秘密共享方案。最后，利用我们的最小码的对偶码得到了民主秘密共享方案。
英文摘要：
Linear codes have diverse applications in secret sharing schemes, secure two-party computation, association schemes, strongly regular graphs, authentication codes and communication. There are a large number of linear codes with few weights in the literature, but a little of them are minimal. In this paper, we are using for the first time weakly regular plateaued balanced functions over the finite fields of odd characteristic in the second generic construction method of linear codes. The main results of this paper are stated below. We first construct several three-weight and four-weight linear codes with flexible parameters from weakly regular plateaued balanced functions. It is worth noting that the (almost) optimal codes may be obtained from these functions. We next observe that all codes obtained in this paper are minimal, thereby they can be directly employed to construct secret sharing schemes with high democracy. Finally, the democratic secret sharing schemes are obtained from the dual codes of our minimal codes.
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