Jacobi polynomials, invariant rings, and generalized $t$-designs

In the present paper, we provide results that relate the Jacobi polynomials in genus $g$. We show that if a code is $t$-homogeneous that is, the codewords of the code for every given weight hold a $t$-design, then its Jacobi polynomial in genus $g$ with composition $T$ with $|T|\leq t$ can be obtained from its weight enumerator in genus~$g$ using the polarization operator. Using this fact, we investigate the invariant ring, which relates the homogeneous Jacobi polynomials of the binary codes in genus $g$. Specifically, the generators of the invariant ring appearing for $g=1$ are obtained. Moreover, we define the split Jacobi polynomials in genus~$g$ and obtain the MacWilliams type identity for it. A split generalization for higher genus cases of the relation between the Jacobi polynomials and weight enumerator of a $t$-homogeneous code also given.