A new bound on the capacity of the binary deletion channel with high deletion probabilities

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) ≥ 0.1185. Fertonani and Duman recently showed that lim sup <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d→1</sub> C(d)/(1-d) ≤ 0.49. In this paper, it is proved that lim <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d→1</sub> C(d)/(1 - d) exists and is equal to inf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> C(d)/(1-d). This result suggests the conjecture that the curve C(d) my be convex in the interval d ∈ [0, 1]. Furthermore, using currently known bounds for C(d), it leads to the upper bound lim <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d→1</sub> C(d)/(1 - d) ≤ 0.4143.