Coding with Feedback
"Correcting a Single Error in Feedback Channels" the problem of correcting a single error with feedback was investigated. It was mainly devoted to binary channels, namely, binary symmetric and asymmetric channels. A general theorem, which allows constructing strategies with one feedback, was proved. For the symmetric channel with one error, it was proved that with two feedbacks one can transmit as many messages as with complete feedback. For the asymmetric channel, some strategies for small lengths have been proposed. Later some results for the binary symmetric channel have been generalized for the q-ary symmetric channel. (These results are not published yet, I can send the draft).
There are multiple ways, how this research can be continued:
- One can try to generalize the results from mentioned paper for the case of 2 errors or a constant number of errors.
- The methods developed in the paper "Correcting a Single Error in Feedback Channels" allow us to construct optimal codes for the asymmetric channel with complete feedback and compute their sizes. However, it requires a lot of time. It would be interesting to prove a general formula for the size of such optimal codes.
- Another direction is to consider non-symmetric q-ary channels.