Deletion channel

A deletion channel is a communications channel model used in coding theory and information theory. In this model, a transmitter sends a bit (a zero or a one), and the receiver either receives the bit (with probability $$p$$) or does not receive anything without being notified that the bit was dropped (with probability $$1-p$$). Determining the capacity of the deletion channel is an open problem.

The deletion channel should not be confused with the binary erasure channel which is much simpler to analyze.

Formal description
Let $$p$$ be the deletion probability, $$0 < p < 1$$. The iid binary deletion channel is defined as follows:

Given an input sequence of $$n$$ bits $$(X_i)$$ as input, each bit in $$X_n$$ can be deleted with probability $$p$$. The deletion positions are unknown to the sender and the receiver. The output sequence $$(Y_i)$$ is the sequence of the $$(X_i)$$ which were not deleted, in the correct order and with no errors.

Capacity
The capacity of the binary deletion channel (as an analytical expression of the deletion rate $$p$$) is unknown. It has a mathematical expression. Several upper and lower bounds are known.