Explicit two-deletion codes with redundancy matching the existential bound

TL;DR

This paper presents explicit binary codes capable of correcting two deletions with sizes matching the theoretical bounds, using augmented Varshamov-Tenengolts codes and providing new list decoding constructions.

Contribution

It introduces explicit constructions of two-deletion correcting codes that match existential bounds and offers the first known list decoding codes for two deletions.

Findings

Codes of size $2^n/n^{4+o(1)}$ correct two deletions.

List decoding codes of size $2^n/n^{3+o(1)}$ with list size two.

Explicit constructions match theoretical bounds up to lower order terms.

Abstract

We give an explicit construction of length-$n$ binary codes capable of correcting the deletion of two bits that have size $2^n/n^{4+o(1)}$. This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size $\Omega(2^n/n^4)$. Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size $\Omega(2^n/n^{3+o(1)})$ that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.