Weak Composition Lattices and Ring-Linear Anticodes

TL;DR

This paper explores the structure of Lee-metric anticodes over certain rings, establishing a lattice framework and introducing new invariants for Lee-metric codes through a novel combinatorial approach.

Contribution

It introduces and characterizes optimal Lee-metric anticodes over p^sZ, forming a lattice structure and establishing a bijection with weak compositions, leading to new code invariants.

Findings

Optimal Lee-metric anticodes form a lattice under inclusion.

A bijection exists between anticodes lattice and weak compositions ordered by dominance.

New invariants for Lee-metric codes are introduced via the anticode approach.

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.