Jacobi polynomials, invariant rings, and generalized $t$-designs
Himadri Shekhar Chakraborty, Nur Hamid, Tsuyoshi Miezaki, Manabu Oura
TL;DR
This work develops a higher-genus framework connecting Jacobi polynomials attached to binary codes with genus-$g$ weight enumerators via a polarization mechanism, extending the classical MacWilliams relations. It proves that for $t$-homogeneous codes, the genus-$g$ Jacobi polynomials $J_{C,T}^{(g)}$ with $|T|\le t$ are obtainable from $W_C^{(g)}$ using the polarization operator $A_{(g)}$ and establishes MacWilliams-type identities for both ordinary and split Jacobi polynomials. The invariant-ring perspective yields explicit generators and Molien-type dimension formulas for genus $1$ and genus $2$ in the Type II setting, with detailed computational confirmations in SageMath up to length $24$ and clear genus-specific generator lists. The paper also introduces split Jacobi polynomials and their MacWilliams identities, broadening the design-theoretic applications and linking to generalized $t$-designs, Assmus–Mattson-type phenomena, and invariant theory under unitary reflection groups.
Abstract
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.