DROPS

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.54

Candidate Tree Codes via Pascal Determinant Cubes

Authors: Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract

Tree codes are combinatorial structures introduced by Schulman [Schulman, 1993] as key ingredients in interactive coding schemes. Asymptotically-good tree codes are long known to exist, yet their explicit construction remains a notoriously hard open problem. Even proposing a plausible construction, without the burden of proof, is difficult and the defining tree code property requires structure that remains elusive. To the best of our knowledge, only one candidate appears in the literature, due to Moore and Schulman [Moore and Schulman, 2014]. We put forth a new candidate for an explicit asymptotically-good tree code. Our construction is an extension of the vanishing rate tree code by Cohen-Haeupler-Schulman [Cohen et al., 2018], and its correctness relies on a conjecture that we introduce on certain Pascal determinants indexed by the points of the Boolean hypercube. Furthermore, using the vanishing distance tree code by Gelles et al. [Gelles et al., 2016] enables us to present a construction that relies on an even weaker assumption. We furnish evidence supporting our conjecture through numerical computation, combinatorial arguments from planar path graphs and based on well-studied heuristics from arithmetic geometry.

Cite as

Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan. Candidate Tree Codes via Pascal Determinant Cubes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2021.54,
  author =	{Ben Yaacov, Inbar and Cohen, Gil and Narayanan, Anand Kumar},
  title =	{{Candidate Tree Codes via Pascal Determinant Cubes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{54:1--54:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  URN =		{urn:nbn:de:0030-drops-147474},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  annote =	{Keywords: Tree codes, Sparse polynomials, Explicit constructions}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2016.47

Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

Authors: Zeyu Guo, Anand Kumar Narayanan, and Chris Umans

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract

The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes ~O(n^{3/2}*log(q)+n*log^2(q)) time to factor polynomials of degree n over the finite field F_q with q elements. A significant open problem is if the 3/2 exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than 3/2 would yield an algorithm for polynomial factorization with exponent better than 3/2.

Cite as

Zeyu Guo, Anand Kumar Narayanan, and Chris Umans. Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{guo_et_al:LIPIcs.MFCS.2016.47,
  author =	{Guo, Zeyu and Narayanan, Anand Kumar and Umans, Chris},
  title =	{{Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.47},
  URN =		{urn:nbn:de:0030-drops-64609},
  doi =		{10.4230/LIPIcs.MFCS.2016.47},
  annote =	{Keywords: algorithms, complexity, finite fields, polynomials, factorization}
}

Search Results

Documents authored by Narayanan, Anand Kumar

Candidate Tree Codes via Pascal Determinant Cubes

Abstract

Cite as

Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

Abstract

Cite as

Thanks for your feedback!

Could not send message