# Guess & Check Codes for Deletions and Synchronization

## TL;DR

This paper introduces Guess & Check (GC) codes, a new family of codes capable of correcting multiple random deletions with high probability, achieving near-optimal redundancy and efficient encoding/decoding, with applications in file synchronization.

## Contribution

The paper presents GC codes, a novel approach that extends deletion correction capabilities to multiple deletions with high probability, using MDS codes and achieving asymptotically optimal redundancy.

## Key findings

- GC codes can correct multiple deletions with high probability.
- GC codes have redundancy of approximately a(delta \, ext{log} \, n)
- The encoding and decoding algorithms are deterministic and polynomial-time.

## Abstract

We consider the problem of constructing codes that can correct $\delta$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions $(\delta=1)$ with an asymptotically optimal redundancy. Finding similar codes for $\delta \geq 2$ deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions $\delta$ occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asymptotically optimal redundancy that is $\Theta(\delta \log n)$. We provide deterministic polynomial time encoding and decoding schemes for these codes. We also describe the applications of GC codes to file synchronization.