# Guess & Check Codes for Deletions, Insertions, and Synchronization

## TL;DR

This paper introduces Guess & Check (GC) codes, a new family of explicit, systematic codes with polynomial-time encoding and decoding, capable of correcting multiple deletions or insertions with high probability, relaxing the zero-error requirement.

## Contribution

The paper presents GC codes that can correct multiple deletions or insertions with high probability, extending beyond the single deletion correction of VT codes, and applies them to file synchronization.

## Key findings

- GC codes correct multiple deletions with high probability.
- GC codes have polynomial-time encoding and decoding algorithms.
- Application demonstrated in file synchronization scenarios.

## 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, dating back to 1965, are zero-error single deletion $(\delta=1)$ correcting codes, and have an asymptotically optimal redundancy. Finding similar codes for $\delta \geq 2$ deletions remains an open problem. In this work, we relax the standard zero-error (i.e., worst-case) decoding requirement by assuming that the positions of the $\delta$ deletions (or insertions) are independent of the codeword. Our contribution is a new family of explicit codes, that we call Guess & Check (GC) codes, that can correct with high probability up to a constant number of $\delta$ deletions (or insertions). GC codes are systematic; and have deterministic polynomial time encoding and decoding algorithms. We also describe the application of GC codes to file synchronization.