Correlated Algebraic-Geometric Codes: Improved List Decoding over Bounded Alphabets

We define a new family of error-correcting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraic-geometric (AG) codes via a (non-obvious) generalization of the approach in the recent breakthrough work of F. Parvaresh and A. Vardy (2005). Our work shows that the PV framework applies to fairly general settings by elucidating the key algebraic concepts underlying it. Also, more importantly, AG codes of arbitrary block length exist over fixed alphabets Sigma, thus enabling us to establish new trade-offs between the list decoding radius and rate over a bounded alphabet size. Similar to algorithms for AG codes from V. Guruswami and M. Sudan (1999, 2001), our encoding/decoding algorithms run in polynomial time assuming a natural polynomial-size representation of the code. For codes based on a specific "optimal" algebraic curve, we also present an expected polynomial time algorithm to construct the requisite representation. This in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for AG codes