Infinite families of $2$-designs from a class of cyclic codes with two non-zeros

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.