Decoding Insertions/Deletions via List Recovery

2025-05-05

AI Summary

This work addresses efficient list decoding of Reed–Solomon (RS) codes over insertion–deletion (insdel) channels. We introduce, for the first time, a reduction from insdel decoding to list recovery—applicable to both adversarial and stochastic (Davey–MacKay) insdel models—and establish a general framework that transforms list-recoverable codes into list-decodable insdel codes. This yields the first polynomial-time adversarial insdel list decoder for RS codes of dimension $k > 2$, tolerating up to $t = O(n/k)$ insdel errors. Extending to random insdel channels, our algorithm succeeds with high probability using list size $n^{1/2+0.001}$, markedly improving the rate–error-correction trade-off. The key conceptual advance is the establishment of list recovery as a new paradigm for insdel decoding, overcoming the long-standing barrier to efficient RS decoding under non-erasure synchronization errors.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Adversarial Search
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 (( ho, 2 ho n + 1, L))-list-recoverable code is a (( ho, 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 (tcdot 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 ( ho), a (( ho, 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.

Problem

• Decoding codes from insertions and deletions efficiently
• Reducing insertion/deletion decoding to list recovery problem
• Applying list-recovery to Reed-Solomon codes for better error correction

Innovation

• Reduces decoding to list recovery problem
• Applies Reed-Solomon codes efficiently
• Adapts Koetter-Vardy algorithm for insertions/deletions