  Combinatorial $t$-designs have been an important research subject for many
years, as they have wide applications in coding theory, cryptography,
communications and statistics. The interplay between coding theory and
$t$-designs has been attracted a lot of attention for both directions. It is
well known that a linear code over any finite field can be derived from the
incidence matrix of a $t$-design, meanwhile, that the supports of all codewords
with a fixed weight in a code also may hold a $t$-design. In this paper, by
determining the weight distribution of a class of linear codes derived from
non-binary Kasami cyclic codes, we obtain infinite families of $2$-designs from
the supports of all codewords with a fixed weight in these codes, and calculate
their parameters explicitly.
