Synchronization Strings: Explicit Constructions, Local Decoding, and Applications

TL;DR

This paper presents new explicit, deterministic constructions of synchronization strings that enable efficient local decoding and have broad applications in error correction and interactive communication over channels with insertions and deletions.

Contribution

It introduces a linear time deterministic construction of synchronization strings and a generalized long-distance version with fast local decoding, advancing error correction and communication protocols.

Findings

Deterministic linear time construction of synchronization strings.

Introduction of long-distance synchronization strings with local decoding.

Development of near linear time insdel error-correcting codes.

Abstract

This paper gives new results for synchronization strings, a powerful combinatorial object that allows to efficiently deal with insertions and deletions in various communication settings: $\bullet$ We give a deterministic, linear time synchronization string construction, improving over an $O(n^5)$ time randomized construction. Independently of this work, a deterministic $O(n\log^2\log n)$ time construction was just put on arXiv by Cheng, Li, and Wu. We also give a deterministic linear time construction of an infinite synchronization string, which was not known to be computable before. Both constructions are highly explicit, i.e., the $i^{th}$ symbol can be computed in $O(\log i)$ time. $\bullet$ This paper also introduces a generalized notion we call long-distance synchronization strings that allow for local and very fast decoding. In particular, only $O(\log^3 n)$ time and access to logarithmically many symbols is required to decode any index. We give several applications for these results: $\bullet$ For any $\delta<1$ and $\epsilon>0$ we provide an insdel correcting code with rate $1-\delta-\epsilon$ which can correct any $O(\delta)$ fraction of insdel errors in $O(n\log^3n)$ time. This near linear computational efficiency is surprising given that we do not even know how to compute the (edit) distance between the decoding input and output in sub-quadratic time. We show that such codes can not only efficiently recover from $\delta$ fraction of insdel errors but, similar to [Schulman, Zuckerman; TransInf'99], also from any $O(\delta/\log n)$ fraction of block transpositions and replications. $\bullet$ We show that highly explicitness and local decoding allow for infinite channel simulations with exponentially smaller memory and decoding time requirements. These simulations can be used to give the first near linear time interactive coding scheme for insdel errors.