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https://doi.org/10.1137/1.9781...
Article . 2013 . Peer-reviewed
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Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets

Authors: Venkatesan Guruswami; Chaoping Xing;

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

Abstract

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 ∊ > 0, we get codes over an alphabet size that can be list decoded up to error fraction 1 – R – ∊ confining close-by messages to a subspace with elements. Previous results for list decoding up to error-fraction 1 – R – ∊ 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. Published version

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Singapore
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Keywords

:Science::Mathematics::Discrete mathematics::Algorithms [DRNTU]

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
11
Top 10%
Average
Top 10%
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bronze