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### 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 f{\"u}r Informatik},
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