2026-09-18T21:22:21Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/386672022-01-13T07:54:29Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Madhu Sudan.
Kopparty, Swastik
Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2007-08-29T20:41:44Z
2007-08-29T20:41:44Z
2006
2007
http://hdl.handle.net/1721.1/38667
163581865
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2007.
Includes bibliographical references (p. 29-31).
We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson and Guruswami-Sudan bounds [Joh62, Joh63, GS99]. In particular, we show that for any ... , there exist arbitrarily large fields ... * Existence: there exists a received word ... that agrees with a super-polynomial number of distinct degree K polynomials on ... points each; * Explicit: there exists a polynomial time constructible received word ... that agrees with a super-polynomial number of distinct degree K polynomials, on ... points each. Ill both cases, our results improve upon the previous state of the art, which was , NM/6 for the existence case [JH01], and a ... for the explicit one [GR,05b]. Furthermore, for 6 close to 1 our bound approaches the Guruswami-Sudan bound (which is ... ) and rules out the possibility of extending their efficient RS list decoding algorithm to any significantly larger decoding radius. Our proof method is surprisingly simple. We work with polynomials that vanish on subspaces of an extension field viewed as a vector space over the base field.
(cont.) These subspace polynomials are a subclass of linearized polynomials that were studied by Ore [Ore33, Ore34] in the 1930s and by coding theorists. For us their main attraction is their sparsity and abundance of roots. We also complement our negative results by giving a list decoding algorithm for linearized polynomials beyond the Johnson-Guruswami-Sudan bounds.
by Swastik Kopparty.
S.M.
31 leaves
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.
http://dspace.mit.edu/handle/1721.1/7582
Electrical Engineering and Computer Science.
Subspace polynomials and list decoding of Reed-Solomon codes
Thesis
application/pdf
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="ffdd3ed2-6293-4aea-909e-cec162066ade">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>eng</Language>
<Title>Subspace polynomials and list decoding of Reed-Solomon codes</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2007</PublicationDate>
<Authors>
<Author>
<DisplayName>Kopparty, Swastik</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>http://dspace.mit.edu/handle/1721.1/7582</License>
<Keyword>Electrical Engineering and Computer Science.</Keyword>
<Abstract>We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson and Guruswami-Sudan bounds [Joh62, Joh63, GS99]. In particular, we show that for any ... , there exist arbitrarily large fields ... * Existence: there exists a received word ... that agrees with a super-polynomial number of distinct degree K polynomials on ... points each; * Explicit: there exists a polynomial time constructible received word ... that agrees with a super-polynomial number of distinct degree K polynomials, on ... points each. Ill both cases, our results improve upon the previous state of the art, which was , NM/6 for the existence case [JH01], and a ... for the explicit one [GR,05b]. Furthermore, for 6 close to 1 our bound approaches the Guruswami-Sudan bound (which is ... ) and rules out the possibility of extending their efficient RS list decoding algorithm to any significantly larger decoding radius. Our proof method is surprisingly simple. We work with polynomials that vanish on subspaces of an extension field viewed as a vector space over the base field.</Abstract>
<Abstract>(cont.) These subspace polynomials are a subclass of linearized polynomials that were studied by Ore [Ore33, Ore34] in the 1930s and by coding theorists. For us their main attraction is their sparsity and abundance of roots. We also complement our negative results by giving a list decoding algorithm for linearized polynomials beyond the Johnson-Guruswami-Sudan bounds.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>