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On Optimal Homogeneous-Metric Codes

Andreas Pyka, Violetta Weger

Abstract

The homogeneous metric can be viewed as a natural extension of the Hamming metric to finite chain rings. It distinguishes between three types of elements: zero, non-zero elements in the socle, and elements outside the socle. Since the Singleton bound is one of the most fundamental and widely studied bounds in classical coding theory, we investigate its analogue for codes over finite chain rings equipped with the homogeneous metric. We provide a complete characterization of Maximum Homogeneous Distance (MHD) codes, showing that MHD codes coincide with lifted MDS codes and are contained within the socle at low rank. Exceptions arise from exceptional MDS codes or single-parity-check codes. We then shift our focus to the Plotkin-type bound in the homogeneous metric and close an existing gap in the theory of constant homogeneous-weight codes by identifying those of minimal length.

On Optimal Homogeneous-Metric Codes

Abstract

The homogeneous metric can be viewed as a natural extension of the Hamming metric to finite chain rings. It distinguishes between three types of elements: zero, non-zero elements in the socle, and elements outside the socle. Since the Singleton bound is one of the most fundamental and widely studied bounds in classical coding theory, we investigate its analogue for codes over finite chain rings equipped with the homogeneous metric. We provide a complete characterization of Maximum Homogeneous Distance (MHD) codes, showing that MHD codes coincide with lifted MDS codes and are contained within the socle at low rank. Exceptions arise from exceptional MDS codes or single-parity-check codes. We then shift our focus to the Plotkin-type bound in the homogeneous metric and close an existing gap in the theory of constant homogeneous-weight codes by identifying those of minimal length.

Paper Structure

This paper contains 7 sections, 21 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum Homogeneous Distance Codes
  4. Plotkin Bound and Constant Homogeneous Weight Codes
  5. Constant weight codes
  6. Asymptotic Bounds
  7. Conclusion

Key Result

Proposition 4

Let $\mathcal{C}$ be an $\mathcal{R}$-linear code of length $n$ and subtype $(k_0, \ldots, k_{s-1})$. Then $\mathcal{C}$ has (up to permutation of columns) a generator matrix in the following standard form where $A_{i,s}\in (\mathcal{R} / \gamma^{s-i} \mathcal{R})^{k_{i}\times (n-K)}$ and $A_{i,j}\in (\mathcal{R} / \gamma^{s-i} \mathcal{R})^{k_i\times k_j}$ for $j< s$. Moreover $\mathcal{C}$ has

Figures (1)

  • Figure 1: Comparison of the asymptotic bounds in the homogeneous metric and the Hamming metric, where $q = 3$

Theorems & Definitions (57)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Definition 5
  • Proposition 6
  • Definition 7
  • Definition 8
  • Lemma 9
  • Theorem 10
  • ...and 47 more