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.
@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}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.4}, URN = {urn:nbn:de:0030-drops-7849}, doi = {10.4230/DagSemProc.06201.4}, annote = {Keywords: Error-correcting codes, localized errors, feedback, variable length codes} }
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