A new bound for the capacity of the deletion channel with high deletion probabilities

TL;DR

This paper investigates the capacity of the binary deletion channel at high deletion probabilities, establishing the existence of a limit for the normalized capacity and proposing a new upper bound of approximately 0.4143.

Contribution

It proves the existence of the limit of C(d)/(1-d) as d approaches 1 and derives a new upper bound, advancing understanding of the channel's capacity at high deletion rates.

Findings

The limit of C(d)/(1-d) exists as d approaches 1.

The limit is equal to the infimum of C(d)/(1-d).

An improved upper bound of approximately 0.4143 is established.

Abstract

Let $C(d)$ be the capacity of the binary deletion channel with deletion probability $d$. It was proved by Drinea and Mitzenmacher that, for all $d$, $C(d)/(1-d)\geq 0.1185 $. Fertonani and Duman recently showed that $\limsup_{d\to 1}C(d)/(1-d)\leq 0.49$. In this paper, it is proved that $\lim_{d\to 1}C(d)/(1-d)$ exists and is equal to $\inf_{d}C(d)/(1-d)$. This result suggests the conjecture that the curve $C(d)$ my be convex in the interval $d\in [0,1]$. Furthermore, using currently known bounds for $C(d)$, it leads to the upper bound $\lim_{d\to 1}C(d)/(1-d)\leq 0.4143$.