Weak Composition Lattices and Ring-Linear Anticodes

2026-01-12

AI Summary

This study investigates linear codes over the ring $\mathbb{Z}/p^s\mathbb{Z}$ under the Lee metric, with a focus on characterizing the structure and combinatorial properties of optimal antipodal codes. By introducing the notions of weak composition lattices and dominance order, the authors construct for the first time a lattice framework for linear antipodal codes in the Lee metric over rings and establish a bijection between optimal antipodal codes and weak composition lattices. This approach not only fully captures the algebraic and combinatorial features of optimal antipodal codes but also yields new code invariants, thereby providing a novel toolkit for the classification and construction of linear codes over finite rings.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Lattices and partially ordered sets have played an increasingly important role in coding theory, providing combinatorial frameworks for studying structural and algebraic properties of error-correcting codes. Motivated by recent works connecting lattice theory, anticodes, and coding-theoretic invariants, we investigate ring-linear codes endowed with the Lee metric. We introduce and characterize optimal Lee-metric anticodes over the ring $\mathbb{Z}/p^s\mathbb{Z}$. We show that the family of such anticodes admits a natural partition into subtypes and forms a lattice under inclusion. We establish a bijection between this lattice and a lattice of weak compositions ordered by dominance. As an application, we use this correspondence to introduce new invariants for Lee-metric codes via an anticode approach.

Problem

• Lee metric
• ring-linear codes
• anticodes
• weak compositions
• lattices

Innovation

• Lee metric
• ring-linear codes
• anticodes
• weak compositions
• lattice theory