Jacobi polynomials and design theory II

Abstract

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.

28 pages