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Weak Composition Lattices and Ring-Linear Anticodes

Jessica Bariffi, Drisana Bhatia, Giuseppe Cotardo, Violetta Weger

TL;DR

The paper develops a lattice- and anticode-based framework for Lee-metric ring-linear codes over the finite chain ring $\mathbb{Z}/p^s\mathbb{Z}$. It introduces a lattice of weak compositions $\Delta_{s+1}(n)$ ordered by dominance and proves a bijection with the lattice of optimal Lee-metric anticodes, enabling a combinatorial classification via subtype and degeneracy. It provides a complete description of optimal Lee-anticodes, including a generator-matrix form and a duality with the anti-isomorphism on the composition lattice. Finally, it defines new invariants for Lee-metric codes, such as $d_r(\mathcal{C})$ and binomial moments, and relates them to generalized Hamming weights, offering tools for code analysis and design in the Lee metric over rings.

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.

Weak Composition Lattices and Ring-Linear Anticodes

TL;DR

The paper develops a lattice- and anticode-based framework for Lee-metric ring-linear codes over the finite chain ring . It introduces a lattice of weak compositions ordered by dominance and proves a bijection with the lattice of optimal Lee-metric anticodes, enabling a combinatorial classification via subtype and degeneracy. It provides a complete description of optimal Lee-anticodes, including a generator-matrix form and a duality with the anti-isomorphism on the composition lattice. Finally, it defines new invariants for Lee-metric codes, such as and binomial moments, and relates them to generalized Hamming weights, offering tools for code analysis and design in the Lee metric over rings.

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 . 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.
Paper Structure (7 sections, 23 theorems, 90 equations, 1 figure)

This paper contains 7 sections, 23 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Lattice of Weak Compositions
  4. Codes and Anticodes
  5. Ring-Linear Codes
  6. Ring-linear Anticodes
  7. Invariants

Key Result

Proposition 2.4

Let $\mathbb{K}$ be a field and let $f:\mathscr{P}\rightarrow\mathbb{K}$ be a function. Define $g:\mathscr{P}\rightarrow\mathbb{K}$ by $g(y)=\sum_{x\leq y}f(x)$ for all $y \in \mathscr{P}$. Then

Figures (1)

  • Figure 1: Graphical representation of the lattice $\Delta_4(3)$.

Theorems & Definitions (80)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Möbius Inversion Formula
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 70 more