On the List Decodability of Insertions and Deletions

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 <inline-formula> <tex-math notation="LaTeX">$\Omega (1)$ </tex-math></inline-formula> that are list-decodable from a 0.707-fraction of insertions. For any <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {I}} \geq 0$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {D}} \in [0,1)$ </tex-math></inline-formula>, there exist <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary codes of rate <inline-formula> <tex-math notation="LaTeX">$\Omega (1)$ </tex-math></inline-formula> that are list-decodable from a <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {I}}$ </tex-math></inline-formula>-fraction of insertions and <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {D}}$ </tex-math></inline-formula>-fraction of deletions, where <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> depends only on <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {I}}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {D}}$ </tex-math></inline-formula>. We also provide efficient encoding and decoding algorithms for list-decoding from <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {I}}$ </tex-math></inline-formula>-fraction of insertions and <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {D}}$ </tex-math></inline-formula>-fraction of deletions for any <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {I}} \geq 0$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\tau _{\mathsf {D}} \in [0,1)$ </tex-math></inline-formula>. Based on the Johnson-type bound, we derive a Plotkin-type upper bound on the code size in the Levenshtein metric.

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