Designs, Codes and Cryptography

, Volume 4, Issue 4, pp 203–211 | Cite as

Maximal three-independent subsets of {0, 1, 2} n

  • A. R. Calderbank
  • P. C. Fishburn


We consider a variant of the classical problem of finding the size of the largest cap in ther-dimensional projective geometry PG(r, 3) over the field IF3 with 3 elements. We study the maximum sizef(n) of a subsetS of IF 3 n with the property that the only solution to the equationx1+x2+x3=0 isx1=x2=x3. Letcn=f(n)1/n andc=sup{c1, c2, ...}. We prove thatc>2.21, improving the previous lower bound of 2.1955 ...


Data Structure Information Theory Discrete Geometry Classical Problem Projective Geometry 
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  1. 1.
    R.C. Bose, Mathematical Theory of the Symmetric Factorial Design,Sankhya, Vol. 8 (1947), pp. 107–166.Google Scholar
  2. 2.
    B. Segré, Le Geometrie di Galois,Ann. Mat. Pura. Appl. Vol. 48 (1959), pp. 1–97.Google Scholar
  3. 3.
    B. Segré, Introduction to Galois Geometries,Atti. Acad. Naz. Lincei Mem., Serie VIII, Vol. VIII (1967).Google Scholar
  4. 4.
    B. Segré, Forme e Geometrie Hermitiane, con Particolare Riguardo al Caso Finito,Ann. Mat. Pura. Appl., Vol. 70 (1965), pp. 1–202.Google Scholar
  5. 5.
    J. McLaughlin, A Simple Group of Order 898, 128, 000,Theory of Finite Groups, Benjamin, New York, (1969), pp. 109–111.Google Scholar
  6. 6.
    R. Hill, On the Largest Size Cap inS 5,3,Rend. Accad. Naz. Lincei, Vol. 8, 54 (1973), pp. 378–384.Google Scholar
  7. 7.
    R. Hill, Caps and Groups,Atti dei Convegni Lincei, Colloquio Internazionale sulle Teorie Combinatorie (Roma 1973), No. 17 (Accad. Maz. Lincei, 1976), pp. 384–394.Google Scholar
  8. 8.
    R. Hill, Caps and Codes,Discrete Math., Vol. 22 (1978), pp. 111–137.Google Scholar
  9. 9.
    A.A. Bruen and J.W.P. Hirschfeld, Applications of Line Geometry Over Finite Fields II. The Hermitian Surface,Geom. Dedicata, Vol. 7 (1978), pp. 333–353.Google Scholar
  10. 10.
    R.A. Games, The Packing Problem for Finite Projective Geometries, Thesis, The Ohio State University, 1980.Google Scholar
  11. 11.
    A.R. Calderbank and W.M. Kantor, The Geometry of Two-Weight Codes,Bull. London Math. Soc., Vol. 18 (1986), pp. 97–122.Google Scholar
  12. 12.
    P. Frankl, R.L. Graham and V. Rödl, On Subsets of Abelian Groups With No 3-Term Arithmetic Progression,J. Combin. Theory Ser. A, Vol. 45 (1987), pp. 157–161.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • A. R. Calderbank
    • 1
  • P. C. Fishburn
    • 1
  1. 1.Mathematical Sciences Research CenterAT&T Bell LaboratoriesMurray Hill

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