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

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
_d→1
C(d)/(1-d) ≤ 0.49. In this paper, it is proved that lim
_d→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 ∈ [0, 1]. Furthermore, using currently known bounds for C(d), it leads to the upper bound lim
_d→1
C(d)/(1 - d) ≤ 0.4143.