Upper bounds on the Rate of Uniformly-Random Codes for the Deletion Channel

TL;DR

This paper establishes tight upper bounds on the maximum rate of uniformly-random codes for the deletion channel, providing insights into their limitations and implications for DNA storage applications.

Contribution

It introduces near-optimal upper bounds on the coding rate for the deletion channel, improving understanding of the limits of uniformly-random codes.

Findings

Upper bounds within 0.1 of known lower bounds for all deletion probabilities.

Simulation results suggest bounds are within 0.05 of the exact rate across all parameters.

Conjecture that positive rates are achievable for all deletion probabilities less than 1.

Abstract

We consider the maximum coding rate achievable by uniformly-random codes for the deletion channel. We prove an upper bound that's within 0.1 of the best known lower bounds for all values of the deletion probability $d,$ and much closer for small and large $d.$ We give simulation results which suggest that our upper bound is within 0.05 of the exact value for all $d$, and within $0.01$ for $d>0.75$. Despite our upper bounds, based on simulations, we conjecture that a positive rate is achievable with uniformly-random codes for all deletion probabilities less than 1. Our results imply impossibility results for the (equivalent) problem of compression of i.i.d. sources correlated via the deletion channel, a relevant model for DNA storage.