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Reconstructing Reed-Solomon Codes from Multiple Noisy Channel Outputs

Shubhransh Singhvi, Han Mao Kiah, Eitan Yaakobi

TL;DR

This paper tackles reconstructing Reed-Solomon codewords from $K$ noisy, independent reads through a $q$-ary discrete memoryless symmetric channel with substitution probability $p$. It extends Koetter–Vardy soft-decision decoding by embedding reliability information from all $K$ reads into a multiplicity matrix, enabling a polynomial-time reconstruction algorithm. The authors derive explicit finite-length and asymptotic rate guarantees that depend only on $(p,K)$, showing reliable recovery when the RS rate $R$ lies below a derived threshold (including a refined KV-based bound), under large blocklength and alphabet size. The work advances probabilistic sequence reconstruction, offering practical, high-performance decoding suitable for DNA storage and related multi-copy communication systems.

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study the problem of efficient reconstruction when each of the $K$ outputs is corrupted by a $q$-ary discrete memoryless symmetric (DMS) substitution channel with substitution probability $p$. Focusing on Reed-Solomon (RS) codes, we adapt the Koetter-Vardy soft-decision decoding algorithm to obtain an efficient reconstruction algorithm. For sufficiently large blocklength and alphabet size, we derive an explicit rate threshold, depending only on $(p, K)$, such that the transmitted codeword can be reconstructed with arbitrarily small probability of error whenever the code rate $R$ lies below this threshold.

Reconstructing Reed-Solomon Codes from Multiple Noisy Channel Outputs

TL;DR

This paper tackles reconstructing Reed-Solomon codewords from noisy, independent reads through a -ary discrete memoryless symmetric channel with substitution probability . It extends Koetter–Vardy soft-decision decoding by embedding reliability information from all reads into a multiplicity matrix, enabling a polynomial-time reconstruction algorithm. The authors derive explicit finite-length and asymptotic rate guarantees that depend only on , showing reliable recovery when the RS rate lies below a derived threshold (including a refined KV-based bound), under large blocklength and alphabet size. The work advances probabilistic sequence reconstruction, offering practical, high-performance decoding suitable for DNA storage and related multi-copy communication systems.

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study the problem of efficient reconstruction when each of the outputs is corrupted by a -ary discrete memoryless symmetric (DMS) substitution channel with substitution probability . Focusing on Reed-Solomon (RS) codes, we adapt the Koetter-Vardy soft-decision decoding algorithm to obtain an efficient reconstruction algorithm. For sufficiently large blocklength and alphabet size, we derive an explicit rate threshold, depending only on , such that the transmitted codeword can be reconstructed with arbitrarily small probability of error whenever the code rate lies below this threshold.
Paper Structure (8 sections, 13 theorems, 71 equations, 1 figure, 2 algorithms)

This paper contains 8 sections, 13 theorems, 71 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prior work
  3. Our Contributions
  4. Related Work
  5. Soft Decoding á la Koetter and Vardy
  6. Sequence Reconstruction over the $K$-Draw Discrete Memoryless Symmetric Channel
  7. RS Codes over the $\mathsf{DMS}_{q,K}(p)$ Channel
  8. Conclusion

Key Result

Theorem 1

Let ${\cal C}$ be an $[n,k]_q$ RS code defined over evaluation points $\bm{\alpha} = (\alpha_1,\ldots,\alpha_n)$, and let $M$ be a $q \times n$ multiplicity matrix. Given $M$ as input, the KV algorithm outputs a list ${\cal L}$ with the following properties:

Figures (1)

  • Figure 1: Comparison of achievable rate regions of Algorithm \ref{['alg:read-rec-K-reads']} (Theorem \ref{['thm:R_upBound_DMS']}), hard-decision majority-decoder (Corollary \ref{['cor:majority']}) and channel capacity (Theorem \ref{['thm:DMSqKpCapacity']}) for $K=2$, $K=3$ and $K=4$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2: KV03
  • Definition 3: KV03
  • Definition 4: KV03
  • Theorem 1: KV03
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • Corollary 1
  • ...and 13 more