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.[1][2]

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

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 $p$ ) is unknown. It has a mathematical expression. Several upper and lower bounds are known.

References

Mitzenmacher, Michael (2009), "A survey of results for deletion channels and related synchronization channels", Probability Surveys, 6: 1–33, doi:10.1214/08-PS141, MR 2525669.
Kanoria, Yashodhan; Montanari, Andrea (2013), "Optimal coding for the binary deletion channel with small deletion probability", IEEE Transactions on Information Theory, 59 (10): 6192–6219, doi:10.1109/TIT.2013.2262020, MR 3106824.

External links

Implementation of correction for deletion channel

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[1] Mitzenmacher, Michael (2009), "A survey of results for deletion channels and related synchronization channels", Probability Surveys, 6: 1–33, doi:10.1214/08-PS141, MR 2525669.

[2] Kanoria, Yashodhan; Montanari, Andrea (2013), "Optimal coding for the binary deletion channel with small deletion probability", IEEE Transactions on Information Theory, 59 (10): 6192–6219, doi:10.1109/TIT.2013.2262020, MR 3106824.