Chinese remaindering with errors
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Finding smooth integers in short intervals using CRT decoding
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
"Soft-decision" decoding of Chinese remainder codes
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
List decoding of algebraic-geometric codes
IEEE Transactions on Information Theory
Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory
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We define number-theoretic error-correcting codes based on algebraic number fields, thereby providing a generalization of Chinese Remainder Codes akin to the generalization of Reed-Solomon codes to Algebraic-geometric codes. Our construction is very similar to (and in fact less general than) the one given by Lenstra [8], but the parallel with the function field case is more apparent, since we only use the non-archimedean places for the encoding. We prove that over an alphabet size as small as 19, there even exist asymptotically good number field codes of the type we consider. This result is based on the existence of certain number fields that have an infinite class field tower in which some primes of small norm split completely.