On the List Decodability of Insertions and Deletions

TL;DR

This paper establishes bounds and algorithms for list decoding of codes affected by insertions and deletions, showing that certain codes can reliably correct high fractions of such errors.

Contribution

It introduces a Johnson-type upper bound for list decoding with insertions and deletions, and provides efficient encoding and decoding algorithms for these error types.

Findings

Binary codes can be list-decoded from 0.707 fraction of insertions.

Existence of q-ary codes with high rate decodable from specified insertion and deletion fractions.

Derived a Plotkin-type upper bound on code size in the Levenshtein metric.

Abstract

In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $\Omega(1)$ that are list-decodable from a $0.707$-fraction of insertions. For any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$, there exist $q$-ary codes of rate $\Omega(1)$ that are list-decodable from a $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions, where $q$ depends only on $\tau_\mathsf{I}$ and $\tau_\mathsf{D}$. We also provide efficient encoding and decoding algorithms for list-decoding from $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions for any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$. Based on the Johnson-type bound, we derive a Plotkin-type upper bound on the code size in the Levenshtein metric.