Explicit constructions of graphs without short cycles and low density codes

Abstract

We give an explicit construction of regular graphs of degree 2r withn vertices and girth ≧c logn/logr. We use Cayley graphs of factor groups of free subgroups of the modular group. An application to low density codes is given.

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References

  1. [1]

    P. Erdős andH. Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl,Wiss. Z. Univ. Halle—Wittenberg, Math.-Nat. R. 12 (1963), 251–258.

    Google Scholar 

  2. [2]

    R. G. Gallager,Low-density parity-check codes, M. I. T. Press, Cambridge, Mass. 1963.

    Google Scholar 

  3. [3]

    K. C. Gunning,Lectures on modular forms, Ann. Math. Studies48, Princeton University Press, Princeton N. J. 1962.

    MATH  Google Scholar 

  4. [4]

    M. Hall, Jr.,The theory of groups, Macmillan, N. Y. 1959.

  5. [5]

    G. H. Hardy andE. M. Wright,An introduction to the theory of numbers, 5th ed., Clarendon Press, Oxford 1979.

    MATH  Google Scholar 

  6. [6]

    I. Kra,Automorphic forms and Kleinian groups, Benjamin, Reading, Mass. 1972.

    MATH  Google Scholar 

  7. [7]

    W. Magnus, A. Karrass andD. Solitar,Combinatorial group theory, Interscience, N. Y. 1966.

  8. [8]

    H. Walther andH.-J. Voss,Über Kreise in Graphen, VEB Deutscher Verlag der Wiss., Berlin 1974.

    MATH  Google Scholar 

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Margulis, G.A. Explicit constructions of graphs without short cycles and low density codes. Combinatorica 2, 71–78 (1982). https://doi.org/10.1007/BF02579283

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AMS subject classification (1980)

  • 05 C 35
  • 94 B 05
  • 05 C 38
  • 05 C 25
  • 20 E 05
  • 30 F 40