Correlated algebraic-geometric codes: Improved list decoding over bounded alphabets

Abstract

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.

Error correction up to the information-theoretic limit

“…Further work on extending the results in this article to to the framework of algebraic-geometric codes has been done in [7,5]. A surprising application of the ideas in the Parvaresh-Vardy list decoder is the construction of randomness extractors by Guruswami, Umans and Vadhan [11].…”

Les codes algébriques principaux et leur décodage

“…En fait ils atteignent la capacité du décodage en liste τ n ≈ 1 − k n − , mais ces codes ont un alphabet qui croit très vite en fonction de la longueur, de manière exponentielle en 1 2 . Comme toujours les codes géométriques peuvent être utilisés en remplacement des codes de Reed-Solomon dans cette construction pour réduire la croissance de la taille de l'alphabet [23].…”

Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate

“…Specifically, recent progress in algebraic coding theory [Parvaresh and Vardy 2005;Guruswami and Rudra 2008] has led to the construction of explicit codes over large alphabets that achieve the optimal rate versus error correction radius trade-off -namely, they admit efficient list decoding algorithms to correct close to the optimal fraction 1 − R of errors with rate R. List decoding is an error correction model where the f (T ). An extension of the Parvaresh-Vardy codes [2005] to arbitrary AG codes was achieved in [Guruswami and Patthak 2008]. But in these codes, there is a substantial loss in rate since the encoding includes the evaluations of additional function(s) explicitly picked to satisfy a low-degree relation over some residue field.…”

“…Computing such a basis remains an interesting challenge in computational function field theory. Our description and analysis of the list decoding algorithm in this work is self-contained, though it builds strongly on the framework of the algorithms in [Sudan 1997;Parvaresh and Vardy 2005;Guruswami and Patthak 2008;Guruswami and Rudra 2008].…”