  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.
