  In this paper, we introduce some new polynomials associated to linear codes
over $\mathbb{F}_{q}$. In particular, we introduce the notion of split complete
Jacobi polynomials attached to multiple sets of coordinate places of a linear
code over $\mathbb{F}_{q}$, and give the MacWilliams type identity for it. We
also give the notion of generalized $q$-colored $t$-designs. As an application
of the generalized $q$-colored $t$-designs, we derive a formula that obtains
the split complete Jacobi polynomials of a linear code over
$\mathbb{F}_{q}$.Moreover, we define the concept of colored packing (resp.
covering) designs. Finally, we give some coding theoretical applications of the
colored designs for Type~III and Type~IV codes.
