Abstract. 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 (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and 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 Σ, thus enabling us to establish new trade-offs between the list decoding radius and rate over a bounded alphabet size.The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected) over large alphabets. A similar extension of this work along the lines of Guruswami and Rudra could have substantial impact. Indeed, it could give better trade-offs than currently known over a fixed alphabet (say, GF(2 12 )), which in turn, upon concatenation with a fixed, well-understood binary code, could take us closer to the list decoding capacity for binary codes. This may also be a promising way to address the significant complexity drawback of the result of Guruswami and Rudra, and to enable approaching capacity with bounded list size independent of the block length (the list size and decoding complexity in their work are both n Ω(1/ε) where ε is the distance to capacity).Similar to algorithms for AG codes from Sudan (1999) and, 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.

Abstract. 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 (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and 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 Σ, thus enabling us to establish new trade-offs between the list decoding radius and rate over a bounded alphabet size.The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected) over large alphabets. A similar extension of this work along the lines of Guruswami and Rudra could have substantial impact. Indeed, it could give better trade-offs than currently known over a fixed alphabet (say, GF(2 12 )), which in turn, upon concatenation with a fixed, well-understood binary code, could take us closer to the list decoding capacity for binary codes. This may also be a promising way to address the significant complexity drawback of the result of Guruswami and Rudra, and to enable approaching capacity with bounded list size independent of the block length (the list size and decoding complexity in their work are both n Ω(1/ε) where ε is the distance to capacity).Similar to algorithms for AG codes from Sudan (1999) and, 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.

ABSTRACT:We present an efficient randomized algorithm to test if a given function f : F n p → F p (where p is a prime) is a low-degree polynomial. This gives a local test for Generalized Reed-Muller codes over prime fields. For a given integer t and a given real > 0, the algorithm queries f at O(p−1 +1 ) points to determine whether f can be described by a polynomial of degree at most t. If f is indeed a polynomial of degree at most t, our algorithm always accepts, and if f has a relative distance at least from every degree t polynomial, then our algorithm rejects f with probability at least 1 2 . Our result is almost optimal since any such algorithm must query f on at least ( 1 + p t+1 p−1 ) points.

No abstract