Abstract
We give the first known direct construction for linear families of bounded concentrators. The construction is explicit and the results are simple natural bounded concentrators.
Let\(\mathbb{F}_q \) be the field withq elements,g(x)∈F q [x] of degree greater than or equal to 2,\(H = PGL_2 (\mathbb{F}_q )[x]/g(x)\mathbb{F}_q [x]),{\text{ }}B = PGL_2 (\mathbb{F}_q )\) and\(A = \left\{ {\left. {\left( {\begin{array}{*{20}c} a & {b + cx} \\ 0 & 1 \\ \end{array} } \right)} \right|a \in \mathbb{F}_q^* ;b,c \in \mathbb{F}_q } \right\}\). LetI nputs=H/A,O utputs=H/B, and draw an edge betweenaA andbB iffaA∩bB≠ϕ. We prove that for everyq≥5 this graph is an\(\left( {\left| {H/A} \right|,\frac{q}{{q + 1}},q + 1,\frac{{q - 4}}{{q - 3}}} \right)\) concentrator.
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References
N. Alon, Z. Galil, andV. Milman: Better expanders and superconcentrators,J. of Alg. 8 (1987), 337–347.
L. A. Bassalygo: Asymptotically optimal switching circuits,Problems Information Transmission 17 (1981), 206–211.
V. G. Drinfeld: The proof of Peterson's Conjecture forGL(2) over global field of characteristicp, Functional Analysis and its Applications 22 (1988), 28–43.
I. Efrat: Automorphic spectra on the tree ofPGL 2,Enseign. Math. 37, (2) (1991), 31–34.
S. Gelbart:Automorphic Forms on Adele Groups, Princeton University Press, Princeton 1975.
O. Gaber, andZ. Galil: Explicit construction of linear sized superconcentrators,J. of Comp Sys. Sci. 22 (1981), 407–420.
I. M. Gelfand, M. I. Graev, andI. I. Pyatetskii-Shapiro:Representation Theory and Automorphic Functions, W. B. Saunders Com., 1969.
D. Gorenstein:Finite Groups, Chelsea, 1980.
M. Klawe: Limitations on explicit constructions of expanding graphs,SIAM J. Comp. 13 (1984) 155–156.
S. Lang: SL2(R), Springer-Verlag, New-York, 1985.
A. Lubotzky, R. Phillips, andP. Sarnak: Ramanujan graphs,Combinatorica 8(3) 1988, 261–277.
A. Lubotzky:Discrete Groups, Expanding Graphs and Invariant Measures, Birkhauser Progress in Math, 1994.
G. A. Margulis: Explicit construction of concentrators,Problems of Inform. Transmission (1975), 325–332.
M. Morgenstern: Ramanujan diagrams,SIAM J. of Discrete Math., November 1994.
M. Morgenstern: Existence and explicit construction ofq+1 regular Ramanujan graphs for every prime powerq, J. Combinatorial Theory, Series B,62 (1) (1994), 44–62.
M. Morgenstern: Ramanujan Diagrams and Explicit Construction of Expanding Graphs,Ph.D. Thesis, Hebrew Univ. of Jerusalem, 1990.
G. Prasad: Strong approximation for semi-simple groups over function fields,Ann. of Math. 105 (1977) 553–572.
J. P. Serre:Trees, Springer-Verlag, 1980.
A. Siegel: On universal classes of fast high performance hash functions, their time-space tradeoff, and their applications,30th Annual IEEE conference on Foundations of Computer Science, (1989), 20–25.
R. M. Tanner: Explicit concentrators from generalizedn-gons,SIAM J. of Alg. Disc. Math. 5 (1984), 287–294.
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Part of this research was done while the author was at the department of Computer Science, The University of British Columbia, Vancouver, B.C., Canada.
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Morgenstern, M. Natural bounded concentrators. Combinatorica 15, 111–122 (1995). https://doi.org/10.1007/BF01294463
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DOI: https://doi.org/10.1007/BF01294463