Algorithms for $q$-ary Error-Correcting Codes with Limited Magnitude and Feedback

TL;DR

This paper determines the capacity error function for all q-ary wraparound channels with limited magnitude and noiseless feedback, extending binary results and connecting to Shannon's zero-error problem.

Contribution

It provides a complete characterization of the capacity error function for q-ary channels with limited magnitude and feedback, a problem previously unresolved.

Findings

Capacity error function fully characterized for all q-ary wraparound channels with level r

Algorithms utilize partial noiseless feedback for optimal performance

Connects to Shannon's zero-error problem in a special case

Abstract

Berlekamp and Zigangirov completely determined the capacity error function for binary error correcting codes with noiseless feedback. It is still an unsolved problem if the upper bound for the capacity error function in the non-binary case of Ahlswede, Lebedev, and Deppe is sharp. We consider wraparound channels with limited magnitude and noiseless feedback. We completely determine the capacity error function for all $q$-ary wraparound channels with a magnitude of level $r$. All of our algorithms use partial noiseless feedback. Furthermore, a special case of the problem is equivalent to Shannon's zero-error problem.