Combinatorial $t$-designs have wide applications in coding theory, cryptography, communications and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$-design. In this paper, we first determine the weight distribution of a class of linear codes derived from the dual of extended cyclic code with two non-zeros. We then obtain infinite families of $2$-designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By simple counting argument, we obtain exponentially many $2$-designs.

中文翻译：

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来自一类循环码的 2-设计的无限族

组合 $t$-designs 在编码理论、密码学、通信和统计中有着广泛的应用。众所周知，代码中所有具有固定权重的码字的支持可能会给出$t$-design。在本文中，我们首先确定了由两个非零扩展循环码的对偶导出的一类线性码的权重分布。然后我们获得无限的$2$-设计族，并从代码中具有固定权重的所有代码字的支持中明确计算它们的参数。通过简单的计数论证，我们得到成倍多的 $2$-设计。