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Transactions of the American Mathematical Society

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Subspace designs based on algebraic function fields


Authors: Venkatesan Guruswami, Chaoping Xing and Chen Yuan
Journal: Trans. Amer. Math. Soc. 370 (2018), 8757-8775
MSC (2010): Primary 05B30, 11Z05, 11R60, 11T71
DOI: https://doi.org/10.1090/tran/7369
Published electronically: July 31, 2018
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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 (STOC'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 (FOCS'13, Combinatorica '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 \ge m$). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree ``dimension expanders'' 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
Email: guruswami@cmu.edu

Chaoping Xing
Affiliation: School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637616
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
Email: Chen.Yuan@cwi.nl

DOI: https://doi.org/10.1090/tran/7369
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)
Article copyright: © Copyright 2018 American Mathematical Society

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