Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial t-designs have attracted lots of research interest for decades. The interplay between coding theory and t-designs started many years ago. It is generally known that t-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a t-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of 2-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of 2-designs are calculated explicitly.

中文翻译：

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与 Dembowski-Ostrom 函数相关的一类线性码的无限二设计族

由于它们在编码理论、密码学、通信和统计学中的重要应用，组合吨几十年来，设计吸引了很多研究兴趣。编码理论与吨-设计始于多年前。众所周知，吨-设计可用于在任何有限域上推导线性代码，并且代码中具有固定权重的所有代码字的支持也可以保持吨-设计。在本文中，我们首先从与 Dembowski-Ostrom 函数相关的循环码构造了一类线性码。通过使用指数和，我们然后确定线性码的权重分布。最后，我们获得了无限的家庭2-根据这些代码中具有固定权重的所有代码字的支持设计。此外，参数2-设计是明确计算的。