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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Subspace designs based on algebraic function fields
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by Venkatesan Guruswami, Chaoping Xing and Chen Yuan PDF
Trans. Amer. Math. Soc. 370 (2018), 8757-8775 Request permission

Abstract:

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\mathbb {F}_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The notion was put forth by Guruswami and Xing (STOCā€™13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes and later also used for explicit rank-metric codes with optimal list decoding radius.

Guruswami and Kopparty (FOCSā€™13, Combinatorica ā€™16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes and required large field size (specifically $q \geqslant m$). Forbes and Guruswami (RANDOMā€™15) used this construction to give explicit constant degree ā€œdimension expandersā€ over large fields and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness.

Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over $\mathbb {F}^n$ for any field $\mathbb {F}$, with logarithmic degree and expansion guarantee for subspaces of dimension $\Omega (n/(\log \log n))$.

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Additional Information
  • Venkatesan Guruswami
  • Affiliation: Computer Science Department, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
  • MR Author ID: 611277
  • Email: guruswami@cmu.edu
  • Chaoping Xing
  • Affiliation: School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637616
  • MR Author ID: 264368
  • Email: xingcp@ntu.edu.sg
  • Chen Yuan
  • Affiliation: School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637616
  • Address at time of publication: CWI, Science Park 123, 1098 XG Amsterdam, Netherlands
  • MR Author ID: 910761
  • Email: Chen.Yuan@cwi.nl
  • Received by editor(s): April 23, 2017
  • Received by editor(s) in revised form: August 1, 2017
  • Published electronically: July 31, 2018
  • Additional Notes: Research of the first author is supported in part by NSF grants CCF-1422045 and CCF-1563742.
    Research of the second author is supported in part by the Singapore MoE Tier 1 grant RG25/16 and NTU Grant M4081575.
    Research of the third author is supported in part by the grant ERC H2020 grant No.74079 (ALGSTRONGCRYPTO)
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 8757-8775
  • MSC (2010): Primary 05B30, 11Z05, 11R60, 11T71
  • DOI: https://doi.org/10.1090/tran/7369
  • MathSciNet review: 3864394