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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.867

Deletion Codes in the High-noise and High-rate Regimes

Authors: Venkatesan Guruswami and Carol Wang

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst-case deletions, with a focus on constructing efficiently encodable and decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following trade-offs (for any epsilon > 0): (1) Codes that can correct a fraction 1-epsilon of deletions with rate poly(eps) over an alphabet of size poly(1/epsilon); (2) Binary codes of rate 1-O~(sqrt(epsilon)) that can correct a fraction eps of deletions; and (3) Binary codes that can be list decoded from a fraction (1/2-epsilon) of deletions with rate poly(epsion) Our work is the first to achieve the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate aproaching 1. The above results bring our understanding of deletion code constructions in these regimes to a similar level as worst-case errors.

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Venkatesan Guruswami and Carol Wang. Deletion Codes in the High-noise and High-rate Regimes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 867-880, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2015.867,
  author =	{Guruswami, Venkatesan and Wang, Carol},
  title =	{{Deletion Codes in the High-noise and High-rate Regimes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{867--880},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.867},
  URN =		{urn:nbn:de:0030-drops-53417},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.867},
  annote =	{Keywords: algorithmic coding theory, deletion codes, list decoding, probabilistic method, explicit constructions}
}

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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.748

Evading Subspaces Over Large Fields and Explicit List-decodable Rank-metric Codes

Authors: Venkatesan Guruswami and Carol Wang

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract

We construct an explicit family of linear rank-metric codes over any field F that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1-rho-eps, for any desired rho in (0,1) and eps > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an F-subspace that evades certain structured subspaces over an extension field of F. These structured spaces arise from the linear-algebraic list decoder for Gabidulin codes due to Guruswami and Xing (STOC'13). Our construction is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which are a collection of subspaces, every pair of which have low-dimensional intersection, and which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order that are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list decoding RS codes reduces to list decoding such folded RS codes. However, as we only list decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list decoding RS codes.

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Venkatesan Guruswami and Carol Wang. Evading Subspaces Over Large Fields and Explicit List-decodable Rank-metric Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 748-761, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2014.748,
  author =	{Guruswami, Venkatesan and Wang, Carol},
  title =	{{Evading Subspaces Over Large Fields and Explicit List-decodable Rank-metric Codes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{748--761},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.748},
  URN =		{urn:nbn:de:0030-drops-47361},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.748},
  annote =	{Keywords: list-decoding, pseudorandomness, algebraic coding, explicit constructions}
}

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Documents authored by Wang, Carol

Deletion Codes in the High-noise and High-rate Regimes

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Evading Subspaces Over Large Fields and Explicit List-decodable Rank-metric Codes

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