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In this paper, we introduce some new polynomials associated to linear codesover $\mathbb{F}_{q}$. In particular, we introduce the notion of split completeJacobi polynomials attached to multiple sets of coordinate places of a linearcode over $\mathbb{F}_{q}$, and give the MacWilliams type identity for it. Wealso give the notion of generalized $q$-colored $t$-designs. As an applicationof the generalized $q$-colored $t$-designs, we derive a formula that obtainsthe 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 thecolored designs for Type~III and Type~IV codes.

Jacobi polynomials and design theory II