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Algebraic codes that achieve list decoding capacity were recently constructedby a careful ``folding'' of the Reed-Solomon code. The ``low-degree'' nature ofthis folding operation was crucial to the list decoding algorithm. We show howsuch folding schemes conducive to list decoding arise out of theArtin-Frobenius automorphism at primes in Galois extensions. Using thisapproach, we construct new folded algebraic-geometric codes for list decodingbased on cyclotomic function fields with a cyclic Galois group. Such functionfields are obtained by adjoining torsion points of the Carlitz action of anirreducible $M \in \F_q[T]$. The Reed-Solomon case corresponds to the simplestsuch extension (corresponding to the case $M=T$). In the general case, we needto descend to the fixed field of a suitable Galois subgroup in order to ensurethe existence of many degree one places that can be used for encoding.  Our methods shed new light on algebraic codes and their list decoding, andlead to new codes achieving list decoding capacity. Quantitatively, these codesprovide list decoding (and list recovery/soft decoding) guarantees similar tofolded Reed-Solomon codes but with an alphabet size that is onlypolylogarithmic in the block length. In comparison, for folded RS codes, thealphabet size is a large polynomial in the block length. This has applicationsto fully explicit (with no brute-force search) binary concatenated codes forlist decoding up to the Zyablov radius.

Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes