In this paper, we introduce some new polynomials associated to linear codes
over Fqβ. In particular, we introduce the notion of split complete
Jacobi polynomials attached to multiple sets of coordinate places of a linear
code over Fqβ, 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
Fqβ.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.Comment: 28 page