# Polar Coding for the Binary Erasure Channel with Deletions

## TL;DR

This paper explores the use of polar codes for binary erasure channels with deletions, proposing a list decoding algorithm with redundancy optimization, achieving high probability message recovery with manageable complexity.

## Contribution

It introduces a polar coding scheme for deletion channels, including a list decoding algorithm with redundancy optimization and complexity analysis.

## Key findings

- Decoding complexity is $O(N^2\log N)$.
- High probability of message recovery as code length increases.
- List size can be reduced to one in simulations.

## Abstract

We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is $O(N^2\log N)$, where $N$ is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations which show that the list size can be reduced to one and the original message can be recovered with high probability as the length of the code grows.