# Synchronization Strings: Efficient and Fast Deterministic Constructions over Small Alphabets

## TL;DR

This paper improves bounds on the minimal alphabet size for synchronization strings, provides efficient explicit constructions over small alphabets, and explores the extremal question of their existence over various alphabet sizes.

## Contribution

It offers new lower and upper bounds on alphabet size for synchronization strings and presents fast explicit constructions over small alphabets.

## Key findings

- Synchronization strings can be constructed over alphabets of size four.
- No synchronization strings exist over binary alphabets for certain parameters.
- Open problem remains on existence over ternary alphabets.

## Abstract

Synchronization strings are recently introduced by Haeupler and Shahrasbi (STOC 2017) in the study of codes for correcting insertion and deletion errors (insdel codes). They showed that for any parameter $\varepsilon>0$, synchronization strings of arbitrary length exist over an alphabet whose size depends only on $\varepsilon$. Specifically, they obtained an alphabet size of $O(\varepsilon^{-4})$, which left an open question on where the minimal size of such alphabets lies between $\Omega(\varepsilon^{-1})$ and $O(\varepsilon^{-4})$. In this work, we partially bridge this gap by providing an improved lower bound of $\Omega(\varepsilon^{-3/2})$, and an improved upper bound of $O(\varepsilon^{-2})$. We also provide fast explicit constructions of synchronization strings over small alphabets. Further, along the lines of previous work on similar combinatorial objects, we study the extremal question of the smallest possible alphabet size over which synchronization strings can exist for some constant $\varepsilon < 1$. We show that one can construct $\varepsilon$-synchronization strings over alphabets of size four while no such string exists over binary alphabets. This reduces the extremal question to whether synchronization strings exist over ternary alphabets.