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Weighted-Hamming Metric for Parallel Channels

Sebastian Bitzer, Alberto Ravagnani, Violetta Weger

TL;DR

The paper studies a weighted-Hamming metric tailored for independent parallel $q$-ary symmetric channels, showing that maximum-likelihood decoding can be achieved by minimizing a weighted distance with scaling factors determined by subchannel reliabilities. It extends classical coding bounds to the weighted-Hamming setting, including Singleton-like, Plotkin-like, Hamming-like, Gilbert-Varshamov, and LP bounds, and demonstrates that the metric is not normal, enabling error-correction beyond half the minimum distance in some codes. A simple construction achieves optimal minimum distance for specific parameter regimes, illustrating practical MWHD code design. The results provide both theoretical insight and practical guidance for reliable worst-case performance in parallel-channel communications, with potential extensions to polyalphabetic codes as future work.

Abstract

Independent parallel q-ary symmetric channels are a suitable transmission model for several applications. The proposed weighted-Hamming metric is tailored to this setting and enables optimal decoding performance. We show that some weighted-Hamming-metric codes exhibit the unusual property that all errors beyond half the minimum distance can be corrected. Nevertheless, a tight relation between the error-correction capability of a code and its minimum distance can be established. Generalizing their Hamming-metric counterparts, upper and lower bounds on the cardinality of a code with a given weighted-Hamming distance are obtained. Finally, we propose a simple code construction with optimal minimum distance for specific parameters.

Weighted-Hamming Metric for Parallel Channels

TL;DR

The paper studies a weighted-Hamming metric tailored for independent parallel -ary symmetric channels, showing that maximum-likelihood decoding can be achieved by minimizing a weighted distance with scaling factors determined by subchannel reliabilities. It extends classical coding bounds to the weighted-Hamming setting, including Singleton-like, Plotkin-like, Hamming-like, Gilbert-Varshamov, and LP bounds, and demonstrates that the metric is not normal, enabling error-correction beyond half the minimum distance in some codes. A simple construction achieves optimal minimum distance for specific parameter regimes, illustrating practical MWHD code design. The results provide both theoretical insight and practical guidance for reliable worst-case performance in parallel-channel communications, with potential extensions to polyalphabetic codes as future work.

Abstract

Independent parallel q-ary symmetric channels are a suitable transmission model for several applications. The proposed weighted-Hamming metric is tailored to this setting and enables optimal decoding performance. We show that some weighted-Hamming-metric codes exhibit the unusual property that all errors beyond half the minimum distance can be corrected. Nevertheless, a tight relation between the error-correction capability of a code and its minimum distance can be established. Generalizing their Hamming-metric counterparts, upper and lower bounds on the cardinality of a code with a given weighted-Hamming distance are obtained. Finally, we propose a simple code construction with optimal minimum distance for specific parameters.
Paper Structure (5 sections, 10 theorems, 41 equations, 1 figure)

This paper contains 5 sections, 10 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Weighted-Hamming Metric
  3. Bounds
  4. Code Construction
  5. Conclusion

Key Result

Theorem 2

Let $\mathbf{r} = (\mathbf r_1,\ldots,\mathbf r_m)$ with $\mathbf r_\ell\in\mathbb{F}_q^{n_\ell}$ be a sequence obtained by transmitting $\mathbf{c} = (\mathbf c_1,\ldots,\mathbf c_m)\in\mathcal{C}$ over $m$ parallel $q$-ary symmetric channels, each with individual error probability $\rho_\ell\in(0,

Figures (1)

  • Figure 1: Bounds on the code size for $\mathbf n=(7,\,7)$, $\bm{\lambda} = (1,\,2)$.

Theorems & Definitions (22)

  • Definition 1: Weighted-Hamming metric
  • Theorem 2
  • proof
  • Example 3
  • Example 4
  • Theorem 5
  • proof
  • Theorem 6: Singleton-like bound
  • proof
  • Theorem 7
  • ...and 12 more