Capacity of the binary deletion channel

Determine the Shannon capacity of the binary deletion channel, i.e., the supremum of achievable rates for reliable communication over a channel where each transmitted bit is independently deleted with a fixed probability.

Background

In discussing prior stochastic models of nanopore sequencing that include synchronization errors (such as dropped k-mers), the authors note that one component of these models is the binary deletion channel. They emphasize that the capacity of this channel remains an open problem, highlighting the broader difficulty of analyzing channels with synchronization errors.

This contextualizes their choice to study a deterministic abstraction without deletions, acknowledging that solving capacity questions for more realistic models would confront the deletion channel’s longstanding open problem.

References

Since understanding the capacity of the binary deletion channel is a difficult open problem, this makes their problem extremely difficult.

— On Coding for an Abstracted Nanopore Channel for DNA Storage  (2102.01839 - Hulett et al., 2021) in Related Work, Section 1.1

The exact capacity of the binary deletion channel remains unknown despite decades of work on achievable rates and converse bounds.

— Binary Deletion Channel Capacity to Within One Hundredth of a Bit  (2609.19412 - Papailiopoulos, 16 Sep 2026) in Abstract; Section 1, Introduction

Because $C(d)$ is unknown, there is no percent error relative to capacity.

— Orbit Reduction and Learned Run Distributions for Finite-Blocklength Binary Deletion Channels  (2609.24889 - Khodaiemehr et al., 21 Sep 2026) in Section 8.1, “Three notions of accuracy” (Section \ref{sec:accuracy})