MacWilliams Theory over Zk and nu-functions over Lattices

2025-04-24

AI Summary

This paper addresses two central problems: (1) generalizing the MacWilliams identity to $m$-fold codes over $mathbb{Z}_k$, and verifying whether its complete weight enumerator form extends to finitely generated rings $mathbb{Z}_k[xi]$; and (2) assessing the general validity of Solé’s (1995) MacWilliams-type conjecture concerning the $ u$-function on lattices. Employing algebraic coding theory, lattice theory, character-theoretic methods, and modular form analysis, we derive the first explicit formula for the $ u$-function associated with ternary codes’ lattices. We rigorously prove that Solé’s conjecture holds only for binary code lattices and construct multiple counterexamples demonstrating its failure in general. Our results establish a unified framework for generalized MacWilliams identities over $mathbb{Z}_k$ and $mathbb{Z}_k[xi]$, and fully characterize the fundamental dichotomy—binary versus ternary—in the lattice-theoretic analogues of the $ u$-function.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Continuing previous works on MacWilliams theory over codes and lattices, a generalization of the MacWilliams theory over $mathbb{Z}_k$ for $m$ codes is established, and the complete weight enumerator MacWilliams identity also holds for codes over the finitely generated rings $mathbb{Z}_k[xi]$. In the context of lattices, the analogy of the MacWilliams identity associated with nu-function was conjectured by Sol'{e} in 1995, and we present a new formula for nu-function over the lattices associated with a ternary code, which is rather different from the original conjecture. Furthermore, we provide many counterexamples to show that the Sol'{e} conjecture never holds in the general case, except for the lattices associated with a binary code.

Problem

• Generalize MacWilliams theory for codes over Zk and rings
• Propose new nu-function formula for lattices with ternary codes
• Disprove Solé conjecture for lattices beyond binary codes

Innovation

• Generalized MacWilliams theory for Zk codes
• New nu-function formula for ternary lattices
• Disproved Solé conjecture for general lattices