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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon, N., Bergmann, E.E., Coppersmith D., Odlyzko, A.M.: Balancing sets of vectors. IEEE Transactions (to appear)
Delsarte, P.: On the four principal parameters of a code. Inf. Control23, 407–438 (1973)
Frankl, P.: Orthogonal vectors in then-dimensional cube and codes with missing distances. Combinatorica6 (1986) (to appear)
Frankl, P., Füredi, Z.: The Erdös-Ko-Rado theorem for integer sequences, SIAM J. Algebraic Discrete Methods1, 376–381 (1980)
Frankl, P., Füredi, Z.: Forbidding just one intersection. J. Comb. Theory (A)39, 160–176 (1985)
Frankl, P., Rödl, V.: Forbidden intersections. Trans. Amer. Math. Soc. (to appear)
Graham, R.L.: Personal communication
Ito, N.: Hadamard graphs I, Graphs and Combinatorics1, 57–64 (1985)
Ito, N.: Hadamard graphs II, Graphs and Combinatorics1, 331–337 (1985)
Kleitman, D.J.: On a combinatorial conjecture of Erdös. J. Comb. Theory1, 209–214 (1966)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math.13, 383–390 (1975)
Larman, D.G., Rogers, C.A.: The realization of distances within sets in Euclidean space. Mathematika19, 1–24 (1972)
Mallows, C.L., Sloane, N.J.A.: Weight enumerators of self-orthogonal codes. Discrete Math.9, 391–400 (1974)
Olson, J.E.: A combinatorial problem on finite abelian groups. J. Number Theory1, 8–10 (1969)
Pless, V.: Introduction to the Theory of Error-Correcting Codes. New York: Wiley 1982
Stein, S.K.: Two combinatorial covering theorems. J. Comb. Theory (A)16, 391–397 (1974)
Ward, H.N.: Divisible codes. Arch. Math.36, 485–494 (1981)
Wilson, R.M.: The exact bound in the Erdös-Ko-Rado theorem. Combinatorica4, 247–257 (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Enomoto, H., Frankl, P., Ito, N. et al. Codes with given distances. Graphs and Combinatorics 3, 25–38 (1987). https://doi.org/10.1007/BF01788526
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01788526