, Volume 15, Issue 1, pp 111–122

Natural bounded concentrators

  • Moshe Morgenstern


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)Fq[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\(A = \left\{ {\left. {\left( {\begin{array}{*{20}c} a & {b + cx} \\ 0 & 1 \\ \end{array} } \right)} \right|a \in \mathbb{F}_q^* ;b,c \in \mathbb{F}_q } \right\}\). LetInputs=H/A,Outputs=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.

Mathematics Subject Classification (1991)

Primary: 05 C 35 Secondary: 05 C 25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Alon, Z. Galil, andV. Milman: Better expanders and superconcentrators,J. of Alg. 8 (1987), 337–347.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    L. A. Bassalygo: Asymptotically optimal switching circuits,Problems Information Transmission 17 (1981), 206–211.MATHGoogle Scholar
  3. [3]
    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.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    I. Efrat: Automorphic spectra on the tree ofPGL 2,Enseign. Math. 37, (2) (1991), 31–34.MATHMathSciNetGoogle Scholar
  5. [5]
    S. Gelbart:Automorphic Forms on Adele Groups, Princeton University Press, Princeton 1975.MATHGoogle Scholar
  6. [6]
    O. Gaber, andZ. Galil: Explicit construction of linear sized superconcentrators,J. of Comp Sys. Sci. 22 (1981), 407–420.CrossRefGoogle Scholar
  7. [7]
    I. M. Gelfand, M. I. Graev, andI. I. Pyatetskii-Shapiro:Representation Theory and Automorphic Functions, W. B. Saunders Com., 1969.Google Scholar
  8. [8]
    D. Gorenstein:Finite Groups, Chelsea, 1980.Google Scholar
  9. [9]
    M. Klawe: Limitations on explicit constructions of expanding graphs,SIAM J. Comp. 13 (1984) 155–156.MathSciNetGoogle Scholar
  10. [10]
    S. Lang: SL2(R), Springer-Verlag, New-York, 1985.Google Scholar
  11. [11]
    A. Lubotzky, R. Phillips, andP. Sarnak: Ramanujan graphs,Combinatorica 8(3) 1988, 261–277.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Lubotzky:Discrete Groups, Expanding Graphs and Invariant Measures, Birkhauser Progress in Math, 1994.Google Scholar
  13. [13]
    G. A. Margulis: Explicit construction of concentrators,Problems of Inform. Transmission (1975), 325–332.Google Scholar
  14. [14]
    M. Morgenstern: Ramanujan diagrams,SIAM J. of Discrete Math., November 1994.Google Scholar
  15. [15]
    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.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    M. Morgenstern: Ramanujan Diagrams and Explicit Construction of Expanding Graphs,Ph.D. Thesis, Hebrew Univ. of Jerusalem, 1990.Google Scholar
  17. [17]
    G. Prasad: Strong approximation for semi-simple groups over function fields,Ann. of Math. 105 (1977) 553–572.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    J. P. Serre:Trees, Springer-Verlag, 1980.Google Scholar
  19. [19]
    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.Google Scholar
  20. [20]
    R. M. Tanner: Explicit concentrators from generalizedn-gons,SIAM J. of Alg. Disc. Math. 5 (1984), 287–294.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 1995

Authors and Affiliations

  • Moshe Morgenstern
    • 1
  1. 1.Department of MathematicsThe Hebrew UniversityJerusalemIsrael

Personalised recommendations