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 ${\displaystyle p}$) or does not receive anything without being notified that the bit was dropped (with probability ${\displaystyle 1-p}$). Determining the capacity of the deletion channel is an open problem.^[1][2]

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

Formal description

Let ${\displaystyle p}$ be the deletion probability, ${\displaystyle 0<p<1}$. The iid binary deletion channel is defined as follows:

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

Capacity

Unsolved problem in computer science

What is the capacity of a deletion channel?

More unsolved problems in computer science

The capacity of the binary deletion channel (as an analytical expression of the deletion rate ${\displaystyle p}$) is unknown. It has a mathematical expression^{[citation needed]}. Several upper and lower bounds are known.