Graphs and Combinatorics

, Volume 3, Issue 1, pp 25–38 | Cite as

Codes with given distances

  • H. Enomoto
  • P. Frankl
  • N. Ito
  • K. Nomura
Article

Abstract

One of the main results says that ifC is a binary linear code of length 4t and of dimension greater than 2t, thenC contains a word of weight 2t and this bound is best possible. Several results of similar flavor are established both for linear and non-linear codes. For the proof a lemma introducing the binormal forms of binary matrices is needed. The results are applied to determine the coset chromatic number of Hadamard graphs, to solve a problem of Galvin and to give a short proof of a theorem of Gleason on self-dual doubly-even codes.

Keywords

Linear Code Chromatic Number Short Proof Binary Matrice Binary Linear Code 

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References

  1. 1.
    Alon, N., Bergmann, E.E., Coppersmith D., Odlyzko, A.M.: Balancing sets of vectors. IEEE Transactions (to appear)Google Scholar
  2. 2.
    Delsarte, P.: On the four principal parameters of a code. Inf. Control23, 407–438 (1973)Google Scholar
  3. 3.
    Frankl, P.: Orthogonal vectors in then-dimensional cube and codes with missing distances. Combinatorica6 (1986) (to appear)Google Scholar
  4. 4.
    Frankl, P., Füredi, Z.: The Erdös-Ko-Rado theorem for integer sequences, SIAM J. Algebraic Discrete Methods1, 376–381 (1980)Google Scholar
  5. 5.
    Frankl, P., Füredi, Z.: Forbidding just one intersection. J. Comb. Theory (A)39, 160–176 (1985)Google Scholar
  6. 6.
    Frankl, P., Rödl, V.: Forbidden intersections. Trans. Amer. Math. Soc. (to appear)Google Scholar
  7. 7.
    Graham, R.L.: Personal communicationGoogle Scholar
  8. 8.
    Ito, N.: Hadamard graphs I, Graphs and Combinatorics1, 57–64 (1985)Google Scholar
  9. 9.
    Ito, N.: Hadamard graphs II, Graphs and Combinatorics1, 331–337 (1985)Google Scholar
  10. 10.
    Kleitman, D.J.: On a combinatorial conjecture of Erdös. J. Comb. Theory1, 209–214 (1966)Google Scholar
  11. 11.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math.13, 383–390 (1975)Google Scholar
  12. 12.
    Larman, D.G., Rogers, C.A.: The realization of distances within sets in Euclidean space. Mathematika19, 1–24 (1972)Google Scholar
  13. 13.
    Mallows, C.L., Sloane, N.J.A.: Weight enumerators of self-orthogonal codes. Discrete Math.9, 391–400 (1974)Google Scholar
  14. 14.
    Olson, J.E.: A combinatorial problem on finite abelian groups. J. Number Theory1, 8–10 (1969)Google Scholar
  15. 15.
    Pless, V.: Introduction to the Theory of Error-Correcting Codes. New York: Wiley 1982Google Scholar
  16. 16.
    Stein, S.K.: Two combinatorial covering theorems. J. Comb. Theory (A)16, 391–397 (1974)Google Scholar
  17. 17.
    Ward, H.N.: Divisible codes. Arch. Math.36, 485–494 (1981)Google Scholar
  18. 18.
    Wilson, R.M.: The exact bound in the Erdös-Ko-Rado theorem. Combinatorica4, 247–257 (1984)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • H. Enomoto
    • 1
  • P. Frankl
    • 2
  • N. Ito
    • 3
  • K. Nomura
    • 4
  1. 1.Department of Information ScienceUniversity of TokyoTokyoJapan
  2. 2.CNRS, Quai Anatole FranceParisFrance
  3. 3.Department of Applied MathematicsKonan UniversityKobeJapan
  4. 4.Department of MathematicsTokyo Ikashika UniversityKonodai, IchikawaJapan

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