  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.
