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

2025-04-17

AI Summary

This paper investigates the construction of lattices from linear codes over $mathbb{Z}_k$ and their connections to theta functions. Methodologically, it systematically generalizes the MacWilliams identity to the ring $mathbb{Z}_k$ for arbitrary positive integers $k$, establishing complete and symmetric weight enumerator (CWE/SWE) identities valid for genus $g$. When $k = p$ is prime, it transcends the classical Lee-weight framework by constructing an exact correspondence between Hamming-weight enumerators and associated lattice theta functions over cyclotomic fields. The main contributions are: (i) a unification and extension of the seminal results of Bannai–Dougherty–Harada–Oura and van der Geer–Hirzebruch; (ii) the establishment of a novel bridge between $mathbb{Z}_k$-coding theory and lattice theta functions; and (iii) the provision of foundational tools for interdisciplinary research at the intersection of algebraic coding theory, modular forms, and number theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Extend MacWilliams identities to codes over ring Z_k
• Relate theta functions to weight enumerators for Z_k
• Generalize results from prime p to arbitrary integer k

Innovation

• Extend MacWilliams identity to ring Z_k
• Link theta functions with weight enumerators
• Generalize results for prime p to Z_k