Capacity of a Binary Deletion Channel

Question

It is well known that a communication channel with a randomly induced 50% bit error rate has zero capacity but determining the capacity of a binary deletion channel is still an open problem. Why wouldn't a channel with 50% of its bits being deleted also be deemed to have zero capacity?

Answer 1

A channel with 90% of its bits deleted is still capable of transmitting information. For example, you can send one bit by sending it a hundred times. You can probably send more bits this way, say by transmitting $0^{100} 1^{100}$ for 0 and $0^{200} 1^{100}$ for 1. This still doesn't show that the capacity is positive, but makes it seem reasonable. This survey by Mitzenmacher claims that the capacity is at least $(1-p)/9$, compared to $1-p$ in the binary erasure channel.

In contrast, the binary channel with 50% errors is completely opaque - the output distribution doesn't depend on the input at all. You can't even send one bit.