Codes over Finite Ring $\mathbb{Z}_k$, MacWilliams Identity and Theta Function

TL;DR

The paper develops a unified theta-function framework for linear codes over the ring $\mathbb{Z}_k$, establishing complete and symmetrized weight enumerator MacWilliams identities via lattice theta-functions. It proves explicit genus-$g$ MacWilliams identities that hold for any positive integer $k$ and extends cyclotomic-field interpretations to prime moduli, linking $W_C$ to theta-series of associated lattices. The main contributions are the construction of $k$ theta functions $A_j(z)$ that encode complete weights, the genus-$g$ MacWilliams identities, and the cyclotomic-field generalization connecting complete weight enumerators with theta functions in higher-degree settings. This work deepens the interplay between coding theory, lattice theory, and modular forms, with potential implications for lattice-based cryptography and high-dimensional weight enumerator analyses in ring-linear codes.

Abstract

In this paper, we study linear codes over $\mathbb{Z}_k$ based on lattices and theta functions. We obtain the complete weight enumerators MacWilliams identity and the symmetrized weight enumerators MacWilliams identity based on the theory of theta function. We extend the main work by Bannai, Dougherty, Harada and Oura to the finite ring $\mathbb{Z}_k$ for any positive integer $k$ and present the complete weight enumerators MacWilliams identity in genus $g$. When $k=p$ is a prime number, we establish the relationship between the theta function of associated lattices over a cyclotomic field and the complete weight enumerators with Hamming weight of codes, which is an analogy of the results by G. Van der Geer and F. Hirzebruch since they showed the identity with the Lee weight enumerators.