Algebraic codes that achieve list decoding capacity were recently constructed by a careful folding'' of the Reed-Solomon code. The low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the Artin-Frobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible @math . The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case @math ). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.
Independent of our work, Huang and Narayanan @cite_20 also consider AG codes constructed from Galois extensions, and observe how automorphisms of large order can be used for folding such codes. To our knowledge, the only instantiation of this approach that improves on folded RS codes is the one based on cyclotomic function fields from our work. As an alternate approach, they also propose a decoding method that works with folding via automorphisms of small order. This involves computing several coefficients of the power series expansion of the message function at a low-degree place. Unfortunately, piecing together these coefficients into a function could lead to an exponential list size bound. The authors suggest a heuristic assumption under which they can show that for a random received word, the expected list size and running time are polynomially bounded.
@cite_20
1637999745
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate @math over an alphabet of size independent of block length that can correct a fraction of @math errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the Garcia-Stichtenoth tower, this yields codes of rate @math over an alphabet of size @math , that can correct a fraction of @math errors.