# Polynomial time decodable codes for the binary deletion channel

## TL;DR

This paper presents an explicit construction of codes that can be efficiently encoded and decoded to correct deletions in the binary deletion channel, achieving a rate proportional to the deletion probability, improving practical applicability.

## Contribution

It introduces a polynomial-time decodable code construction for the binary deletion channel with a rate proportional to (1-p), advancing beyond existence proofs to explicit algorithms.

## Key findings

- Codes with rate c₀(1-p) are explicitly constructed.
- Encoding and decoding algorithms run in polynomial time.
- The construction works for any deletion probability p < 1.

## Abstract

In the random deletion channel, each bit is deleted independently with probability $p$. For the random deletion channel, the existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p < 1$, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate $c_0 (1-p)$ for an absolute constant $c_0 > 0$.