Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets

We construct a new list-decodable family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. The function fields underlying these codes are constructed using class field theory, specifically Drinfeld modules of rank 1, and designed to have an automorphism of large order that is used to “fold” the AG code. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. The recent linear-algebraic approach to list decoding can be applied to our new codes, and crucially, we use the Chebotarev density theorem to establish a polynomial upper bound on the list-size for list decoding up to an error fraction approaching 1 − R where R is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field. Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound — specifically, for any desired rate R ∈ (0, 1) and constant ε > 0, we get codes over an alphabet size (1/ε) ) that can be list decoded up to error fraction 1−R−ε confining close-by messages to a subspace with NO(1/ε2) elements. Previous results for list decoding up to error-fraction 1−R−ε over constant-sized alphabets were either based on concatenation or involved taking a carefully chosen subcode of algebraic-geometric codes. In contrast, our result shows that these folded algebraic-geometric codes themselves have the claimed list decoding property. Further, our methods to get function fields with the properties needed for constructing and decoding the code might be of independent algebraic interest. ∗Computer Science Department, Carnegie Mellon University, Pittsburgh, USA. Research supported in part by a Packard Fellowship and NSF CCF-0963975. Email: [email protected] †Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore. Email: [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 46 (2013)