  The interplay between coding theory and $t$-designs has attracted a lot of
attention for both directions. It is well known that the supports of all
codewords with a fixed weight in a code may hold a $t$-design. In this paper,
by determining the weight distributions of two classes of linear codes, we
derive infinite families of $2$-designs from the supports of codewords with a
fixed weight in these codes, and explicitly obtain their parameters.
