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DOI: 10.4230/LIPIcs.CCC.2018.4
URN: urn:nbn:de:0030-drops-88859
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8885/
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Guruswami, Venkatesan ; Resch, Nicolas ; Xing, Chaoping

Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs

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Abstract

For a vector space F^n over a field F, an (eta,beta)-dimension expander of degree d is a collection of d linear maps Gamma_j : F^n -> F^n such that for every subspace U of F^n of dimension at most eta n, the image of U under all the maps, sum_{j=1}^d Gamma_j(U), has dimension at least beta dim(U). Over a finite field, a random collection of d = O(1) maps Gamma_j offers excellent "lossless" expansion whp: beta ~~ d for eta >= Omega(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor beta = 1+epsilon with constant degree is a non-trivial goal. We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following: - Lossless expansion over large fields; more precisely beta >= (1-epsilon)d and eta >= (1-epsilon)/d with d = O_epsilon(1), when |F| >= Omega(n). - Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely beta >= Omega(delta d) and eta >= Omega(1/(delta d)) with d=O_delta(1), when |F| >= n^{delta}. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Omega(1),1+Omega(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with beta >rsim sqrt{d} over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.

BibTeX - Entry

@InProceedings{guruswami_et_al:LIPIcs:2018:8885,
  author =	{Venkatesan Guruswami and Nicolas Resch and Chaoping Xing},
  title =	{{Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Rocco A. Servedio},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8885},
  URN =		{urn:nbn:de:0030-drops-88859},
  doi =		{10.4230/LIPIcs.CCC.2018.4},
  annote =	{Keywords: Algebraic constructions, coding theory, linear algebra, list-decoding, polynomial method, pseudorandomness}
}

Keywords: Algebraic constructions, coding theory, linear algebra, list-decoding, polynomial method, pseudorandomness
Seminar: 33rd Computational Complexity Conference (CCC 2018)
Issue Date: 2018
Date of publication: 01.06.2018


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