Improved Bounds for Codes Correcting Insertions and Deletions

TL;DR

This paper presents improved upper and lower bounds on the size of codes capable of correcting insertions and deletions, advancing the theoretical understanding of such codes with bounds surpassing previous results.

Contribution

It introduces new bounds on code size for insertion/deletion correction, utilizing asymmetric list decoding properties and extending classical bounds like Elias and MRRW.

Findings

Upper bounds are tighter for large minimum Levenshtein distance.

Asymptotic bounds surpass Elias and MRRW bounds in binary and quaternary cases.

Lower bounds improve upon Levenshtein's bounds but are asymptotically limited.

Abstract

This paper studies the cardinality of codes correcting insertions and deletions. We give improved upper and lower bounds on code size. Our upper bound is obtained by utilizing the asymmetric property of list decoding for insertions and deletions and can be seen as analogous to the Elias bound in the Hamming metric. Our non-asymptotic bound is better than the existing bounds when the minimum Levenshtein distance is relatively large. The asymptotic bound exceeds the Elias and the MRRW bounds adapted from the Hamming-metric bounds for the binary and the quaternary cases. Our lower bound improves on the bound by Levenshtein, but its effect is limited and vanishes asymptotically.