Abstract
In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \(\mathbb{F}^n \) be the n dimensional linear space over the field \(\mathbb{F}\). Find a small (ideally constant) set of linear transformations from \(\mathbb{F}^n \) to itself {A i } i∈I such that for every linear subspace V ⊂ \(\mathbb{F}^n \) of dimension dim(V)<n/2 we have
where α>0 is some constant. In other words, the dimension of the subspace spanned by {A i (V)} i∈I should be at least (1+α)·dim(V). For fields of characteristic zero Lubotzky and Zelmanov [10] completely solved the problem by exhibiting a set of matrices, of size independent of n, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results.
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1.
We give a constant number of matrices that expand the dimension of every subspace of dimension d<n/2 by a factor of (1+1/logn).
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2.
We give a set of O<(logn) matrices with expanding factor of (1+α), for some constant α>0.
Our constructions are algebraic in nature and rely on expanding Cayley graphs for the group ℤ=ℤn and small-diameter Cayley graphs for the group SL2(p).
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References
N. Alon and Y. Roichman: Random cayley graphs and expanders, Random Structures and Algorithms 5(2) (1994), 271–285.
L. Babai, W. M. Kantor and A. Lubotsky: Small-diameter cayley graphs for finite simple groups, Europ. J. Combinatorics 10(6) (1989), 507–522.
B. Barak, R. Impagliazzo, A. Shpilka and A. Wigderson: Unpublished Manuscript, 2004.
J. Bourgain: On the construction of affine extractors, Geometric And Functional Analysis 17(1) (2007), 33–57.
J. Bourgain: Expanders and dimensional expansion, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
Z. Dvir and A. Wigderson: Monotone expanders — constructions and applications, Manuscript, 2009.
A. Gabizon and R. Raz: Deterministic extractors for affine sources over large fields. Combinatorica 28 (2008), 415–440.
Z. Karnin and A. Shpilka: Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in, In Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity, CCC’ 08, pp. 280–291, Washington, DC, USA, 2008. IEEE Computer Society.
A. Lubotzky: Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics. Birkhauser, 1994.
A. Lubotzky and Y. Zelmanov: Dimension expanders, Journal of Algebra 319(2) (2008), 730–738.
A. Lubotzky and A. Żuk: On property (τ), in preperation. http://www.ma.huji.ac.il/~alexlub/BOOKS/On%20property/On%20property.pdf.
F. J. MacWilliams and N. J. A. Sloane: The Theory of Error-Correcting Codes, Part II, North-Holland, 1977.
R. Meshulam and A. Wigderson: Expanders in group algebras, Combinatorica 24(4) (2004), 659–680.
A. Wigderson and D. Xiao: Derandomizing the ahlswede-winter matrix-valued chernoff bound using pessimistic estimators, and applications, Theory of Computing 4(1) (2008), 53–76.
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Dvir, Z., Shpilka, A. Towards dimension expanders over finite fields. Combinatorica 31, 305–320 (2011). https://doi.org/10.1007/s00493-011-2540-8
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DOI: https://doi.org/10.1007/s00493-011-2540-8