Algebraic-geometric generalizations of the Parvaresh-Vardy codes

This paper is concerned with a new family of error-correcting codes based on algebraic curves over finite fields, and list decoding algorithms for them. The basic goal in the subject of list decoding is to construct error-correcting codes C over some alphabet Σ which have good rate R, and at the same every Hamming ball of (relative) radius p has few codewords of C, and moreover these codewords can be found in polynomial time. The trade-off between the rateR and the error-correction radius p is a central one governing list decoding. Traditional “unique decoding” algorithms can achieve p = (1 − R)/2, and this was improved in [7] to p = 1 − √ R through a new list decoding algorithm for Reed-Solomon (RS) codes. For several years, this remained the best known trade-off between rate and list decoding radius. In a recent breakthrough, Parvaresh and Vardy [11] define a variant of RS codes which can be list decoded beyond the 1 − √ R radius for rates R 6 1/16. We generalize the PV framework to algebraic-geometric (AG) codes, of which RS codes are an important special case. This shows that their framework applies to fairly general settings, and also better elucidates the key algebraic concepts underlying the new codes. Moreover, since AG codes of arbitrary block length exist over fixed alphabets Σ, we are able to almost match the trade-off between p andR obtained in [11] over alphabets of constant size. In contrast, the PV codes have alphabet size that is polynomially large in the block length. Similar to algorithms for AG codes from [7, 8], 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 also presents an efficient construction of the representation needed by the list decoding algorithms for AG codes in [7]. ∗Research supported in part by NSF grant CCF-0343672 and a Sloan Research Fellowship. Electronic Colloquium on Computational Complexity, Report No. 132 (2005)