Constructions of codes from number fields

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 (1986), 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.

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