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Weighted-Hamming Metric: Bounds and Codes

Sebastian Bitzer, Alberto Ravagnani, Violetta Weger

TL;DR

This work addresses the weighted-Hamming metric by directly bounding the true error-correction capability $t_{\bm{\lambda}}(\mathcal{C})$, rather than relying on the minimum distance $d_{\bm{\lambda}}(\mathcal{C})$, and shows that codes can correct more errors than half the minimum distance in some cases. It introduces a GCC-based code-construction framework that accommodates polyalphabetic outer codes and per-block inner codes, providing a lower bound ${d}'_{\bm{\lambda}}(\mathcal{C})$ on the minimum weighted distance and a lower bound ${t}'_{\bm{\lambda}}(\mathcal{C})$ on the true error-correction capability. A modified decoder enables efficient recovery up to ${t}'_{\bm{\lambda}}(\mathcal{C})$, with numerical results demonstrating near-optimal performance for short lengths and achieving bounds such as the covering and, in some cases, the LP bound. Overall, the paper delivers tighter, actionable bounds and a versatile, decodable code construction for applications involving parallel channels with varying reliability or importance across blocks.

Abstract

The weighted-Hamming metric generalizes the Hamming metric by assigning different weights to blocks of coordinates. It is well-suited for applications such as coding over independent parallel channels, each of which has a different level of importance or noise. From a coding-theoretic perspective, the actual error-correction capability of a code under this metric can exceed half its minimum distance. In this work, we establish direct bounds on this capability, tightening those obtained via minimum-distance arguments. We also propose a flexible code construction based on generalized concatenation and show that these codes can be efficiently decoded up to a lower bound on the error-correction capability.

Weighted-Hamming Metric: Bounds and Codes

TL;DR

This work addresses the weighted-Hamming metric by directly bounding the true error-correction capability , rather than relying on the minimum distance , and shows that codes can correct more errors than half the minimum distance in some cases. It introduces a GCC-based code-construction framework that accommodates polyalphabetic outer codes and per-block inner codes, providing a lower bound on the minimum weighted distance and a lower bound on the true error-correction capability. A modified decoder enables efficient recovery up to , with numerical results demonstrating near-optimal performance for short lengths and achieving bounds such as the covering and, in some cases, the LP bound. Overall, the paper delivers tighter, actionable bounds and a versatile, decodable code construction for applications involving parallel channels with varying reliability or importance across blocks.

Abstract

The weighted-Hamming metric generalizes the Hamming metric by assigning different weights to blocks of coordinates. It is well-suited for applications such as coding over independent parallel channels, each of which has a different level of importance or noise. From a coding-theoretic perspective, the actual error-correction capability of a code under this metric can exceed half its minimum distance. In this work, we establish direct bounds on this capability, tightening those obtained via minimum-distance arguments. We also propose a flexible code construction based on generalized concatenation and show that these codes can be efficiently decoded up to a lower bound on the error-correction capability.
Paper Structure (9 sections, 7 theorems, 40 equations, 2 figures, 1 algorithm)

This paper contains 9 sections, 7 theorems, 40 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds
  4. Code Construction
  5. Polyalphabetic Codes
  6. Generalized Concatenated Codes
  7. Weighted-Hamming Metric Decoder
  8. Achievable Parameters
  9. Conclusion

Key Result

Lemma 1

Let $\mathbf a,\mathbf b\in\mathbb{F}_q^n$ with $w_{\mathbf n}(\mathbf a) \preceq w_{\mathbf n}(\mathbf b)$. Then, $w_{\bm{\lambda}}(\mathbf a) \leq w_{\bm{\lambda}}(\mathbf b)$ and $t_{\bm{\lambda}}(\mathbf a) \leq t_{\bm{\lambda}}(\mathbf b)$. Both $w_{\bm{\lambda}}(\mathbf a)$ and $t_{\bm{\lambda

Figures (2)

  • Figure 1: Bounds on the maximum dimension $k$ of a linear code with error-correcion capability $t$. Block lengths $\mathbf n=(7,\,7)$ and scaling factors $\bm{\lambda} = (1,\,2)$.
  • Figure 2: Dimension $k$ of \ref{['const:gcc']} for given minimum distance $d$ (upper row) or given error-correction capability $t$ (lower row). Block lengths $\mathbf n=(7,\,7,\,7)$ and scaling factors $\bm{\lambda} = (1,\,2,\,3)$.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Example 7
  • Example 8
  • proof
  • Example 10
  • ...and 6 more