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Abstract:

The Chinese Remainder Theorem states that a positive integer m is uniquely specified by its remainder modulo k relatively prime integers p 1; : : : ; p k, provided m! Q k i=1 p i. Thus the residues of m modulo relatively prime integers p 1! p 2! \Delta \Delta \Delta! pn form a redundant representation of m if m! Q k i=1 p i and k! n. This gives a number-theoretic construction of an "error-correcting code " that has been considered often in the past (see [41, 19, 35]). In this code a "message " (integer) m! Q k i=1 p i is encoded by the list of its residues modulu p 1; : : : ; pn. By the Chinese Remainder Theorem, if a code-word is corrupted in e! n\Gammak 2 coordinates, then there exists a unique integer m whose corresponding code-word differs from the corrupted word in at most e places. Furthermore, Mandelbaum [25, 26] shows how m can be recovered efficiently given the corrupted word, provided that the p i 's are very close to one another. To deal with arbitrary p i 's, we present a variant of his algorithm that runs in almost linear-time and recovers from e! log p1 log p1 +log pn \Delta (n \Gamma k) errors. Our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than n\Gammak

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