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Decoding Insertions/Deletions via List Recovery

Anisha Banerjee, Roni Con, Antonia Wachter-Zeh, Eitan Yaakobi

TL;DR

This work introduces a principled reduction of insertions/deletions decoding to list recovery, proving that any $(\rho,2\rho n+1,L)$-list-recoverable code enables $(\rho,L)$-list decoding for insdels. It yields a polynomial-time insdel decoder for Reed–Solomon codes when $t\cdot k=O(n)$ and extends to probabilistic channels such as the Davey–MacKay model, where certain list-recoverable RS codes achieve improved rate–error tradeoffs. The paper also adapts the Koetter–Vardy soft-decision decoder to insdel channels using a generalized reliability matrix and demonstrates its viability with multi-sequence data, supported by numerical results. Collectively, the results expand efficient insdel decoding beyond synchronization-based methods, with direct relevance to DNA storage and other applications involving drift-prone channels.

Abstract

In this work, we consider the problem of efficient decoding of codes from insertions and deletions. Most of the known efficient codes are codes with synchronization strings which allow one to reduce the problem of decoding insertions and deletions to that of decoding substitution and erasures. Our new approach, presented in this paper, reduces the problem of decoding insertions and deletions to that of list recovery. Specifically, any \((ρ, 2ρn + 1, L)\)-list-recoverable code is a \((ρ, L)\)-list decodable insdel code. As an example, we apply this technique to Reed-Solomon (RS) codes, which are known to have efficient list-recovery algorithms up to the Johnson bound. In the adversarial insdel model, this provides efficient (list) decoding from \(t\) insdel errors, assuming that \(t\cdot k = O(n)\). This is the first efficient insdel decoder for \([n, k]\) RS codes for \(k>2\). Additionally, we explore random insdel models, such as the Davey-MacKay channel, and show that for certain choices of \(ρ\), a \((ρ, n^{1/2+0.001}, L)\)-list-recoverable code of length \(n\) can, with high probability, efficiently list decode the channel output, ensuring that the transmitted codeword is in the output list. In the context of RS codes, this leads to a better rate-error tradeoff for these channels compared to the adversarial case. We also adapt the Koetter-Vardy algorithm, a famous soft-decision list decoding technique for RS codes, to correct insertions and deletions induced by the Davey-MacKay channel.

Decoding Insertions/Deletions via List Recovery

TL;DR

This work introduces a principled reduction of insertions/deletions decoding to list recovery, proving that any -list-recoverable code enables -list decoding for insdels. It yields a polynomial-time insdel decoder for Reed–Solomon codes when and extends to probabilistic channels such as the Davey–MacKay model, where certain list-recoverable RS codes achieve improved rate–error tradeoffs. The paper also adapts the Koetter–Vardy soft-decision decoder to insdel channels using a generalized reliability matrix and demonstrates its viability with multi-sequence data, supported by numerical results. Collectively, the results expand efficient insdel decoding beyond synchronization-based methods, with direct relevance to DNA storage and other applications involving drift-prone channels.

Abstract

In this work, we consider the problem of efficient decoding of codes from insertions and deletions. Most of the known efficient codes are codes with synchronization strings which allow one to reduce the problem of decoding insertions and deletions to that of decoding substitution and erasures. Our new approach, presented in this paper, reduces the problem of decoding insertions and deletions to that of list recovery. Specifically, any \((ρ, 2ρn + 1, L)\)-list-recoverable code is a \((ρ, L)\)-list decodable insdel code. As an example, we apply this technique to Reed-Solomon (RS) codes, which are known to have efficient list-recovery algorithms up to the Johnson bound. In the adversarial insdel model, this provides efficient (list) decoding from insdel errors, assuming that \(t\cdot k = O(n)\). This is the first efficient insdel decoder for RS codes for . Additionally, we explore random insdel models, such as the Davey-MacKay channel, and show that for certain choices of , a \((ρ, n^{1/2+0.001}, L)\)-list-recoverable code of length can, with high probability, efficiently list decode the channel output, ensuring that the transmitted codeword is in the output list. In the context of RS codes, this leads to a better rate-error tradeoff for these channels compared to the adversarial case. We also adapt the Koetter-Vardy algorithm, a famous soft-decision list decoding technique for RS codes, to correct insertions and deletions induced by the Davey-MacKay channel.
Paper Structure (10 sections, 6 theorems, 6 equations, 2 figures, 1 algorithm)

This paper contains 10 sections, 6 theorems, 6 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our Results
  4. Decoding adversarial insdels via list-recovery
  5. Probabilistic Channels
  6. The Deletion Channel
  7. The Davey-MacKay Channel
  8. Koetter-Vardy Algorithm for Insdels
  9. Reliability Matrix for Insdels
  10. Numerical Results

Key Result

Theorem 2.1

Let $\rho \in [0,1]$ and set $\ell = 2\rho n + 1$. for the sake of notations we assume that $\rho n$ is an integer. Assume that $\mathcal{C}$ is a $(\rho, \ell, L)$-list-recoverable code with an algorithm $\mathtt{List}$-$\mathtt{Recover}$ that runs in time $T$. Let ${\boldsymbol y} \in \mathbb{F}_q

Figures (2)

  • Figure 1: Allowed transitions in the insertion and deletion channel daveyReliableCommunicationChannels2001
  • Figure 2: Frame error rates for a $[100,33]$ RS code over field $\mathbb{F}_{101}$, where $M$ denotes the number of received sequences, $P_{\mathrm{d}}$ the deletion probability and $P_{\mathrm{i}}$ the insertion probability.

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • ...and 4 more