The Insertion List-Decoding Capacity and an Improved Bound on the Deletion List-Decoding Capacity

2026-07-04

AI Summary

This work investigates the list-decoding capacity of binary codes under synchronization errors caused by insertions and deletions, with a focus on clarifying capacity bounds in high-error regimes where they remain poorly understood. By constructing random codes via a symmetric two-state Markov chain and combining information-theoretic and combinatorial coding arguments, the authors precisely characterize the list-decoding capacity for any insertion fraction δ ∈ [0,1] as (1+δ)(1−h(δ/(1+δ))), where h(·) denotes the binary entropy function. They further demonstrate that such codes outperform uniformly random codes over insertion channels. Additionally, the study derives a tighter upper bound on the deletion channel capacity, which asymptotically matches the known binary deletion channel capacity 1−h(δ) as δ→0.

Abstract

Informally, the capacity of list-decoding in a given adversarial error model is the largest rate at which we can list-decode with list size polynomial in the block length. The capacity of list-decoding from insertions and deletions is a basic, yet poorly understood, aspect of coding against synchronization errors. For example, when dealing with a $δ>1/2$ fraction of insertions, the best known lower bounds give little more than the fact that the capacity is positive. Beyond that regime we also only have loose bounds, with the lower bounds stemming from the analysis of uniformly random codes. We make progress in our understanding of the limits of list-decoding binary codes from insertions and deletions. We show that the capacity of list-decoding from a $δ$-fraction of insertions is exactly \begin{equation*} (1+δ)\left(1-h\left(\fracδ{1+δ}\right)\right) \end{equation*} for all $δ\in[0,1]$, achieved with high probability by a code sampled according to a symmetric $2$-state Markov chain. Curiously, we complement this by showing that such an approach does not beat uniformly random coding in list-decoding from deletions. We also give an improved upper bound on the capacity of list-decoding from a $δ$-fraction of deletions, showing in particular that it behaves like $1-h(δ)$ when $δ\to 0$. This matches the asymptotic behavior of the capacity of the binary deletion channel for vanishing deletion probability.

Problem

• list-decoding
• insertion errors
• deletion errors
• coding capacity
• synchronization errors

Innovation

• list-decoding capacity
• insertion errors
• deletion errors
• Markov chain codes
• synchronization errors