  Combinatorial $t$-designs have nice applications in coding theory, finite
geometries and several engineering areas. There are two major methods of
constructing $t$-designs. One of them is via group actions of certain
permutation groups which are $t$-transitive or $t$-homogeneous on some point
set. The other is a coding-theoretical one. The objectives of this paper are to
introduce two constructions of $t$-designs with special polynomials over finite
fields GF$(q)$, and obtain $2$-designs and $3$-designs with interesting
parameters. A type of d-polynomials is defined and used to construct
$2$-designs. Under the framework of the first construction, it is shown that
every o-polynomial over GF$(2^m)$ gives a $2$-design, and every o-monomial over
GF$(2^m)$ yields a $3$-design. Under the second construction, every
$o$-polynomial gives a $3$-design. Some open problems and conjectures are also
presented in this paper.
