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### Non--binary error correcting codes with noiseless feedback, localized errors, or both

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### Abstract

We investigate non--binary error correcting codes with noiseless feedback, localized errors, or both. It turns out that the Hamming bound is a central concept. For block codes with feedback we present here a coding scheme based on an idea of erasions, which we call the {\bf rubber method}. It gives an optimal rate for big error correcting fraction $\tau$ ($>{1\over q}$) and infinitely many points on the Hamming bound for small $\tau$.

We also consider variable length codes with all lengths bounded from above by $n$ and the end of a word carries the symbol $\Box$ and is thus recognizable by the decoder. For both, the $\Box$-model with feedback and the $\Box$-model with localized errors, the Hamming bound is the exact capacity curve for $\tau <1/2.$ Somewhat surprisingly, whereas with feedback the capacity curve coincides with the Hamming bound also for
$1/2\leq \tau \leq 1$, in this range for localized errors the capacity curve equals 0.

Also we give constructions for the models with both, feedback and localized errors.

### BibTeX - Entry

@InProceedings{ahlswede_et_al:DagSemProc.06201.4,
author =	{Ahlswede, Rudolf and Deppe, Christian and Lebedev, Vladimir},
title =	{{Non--binary error correcting codes with noiseless feedback, localized errors, or both}},
booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
pages =	{1--4},
series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
ISSN =	{1862-4405},
year =	{2006},
volume =	{6201},
editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
}