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Keywords:

  • group divisible t-designs;
  • transversal designs;
  • asymptotic existence

Abstract

The following result gives the answer to the question of whether for a fixed number of groups u and fixed block size k a group divisible t-design of group type inline image with block size k and index one exists for sufficiently large m if the necessary arithmetic conditions are satisfied. Let k and u be positive integers, inline image. Then there exists an integer inline image such that there exists a group divisible t-design of group type inline image with block size k and index one for any integer inline image satisfying the necessary arithmetic conditions

  • display math

The inline image case of this theorem gives an existence theorem for transversal t-designs of large order, which was previously proved by J. L. Blanchard in an unpublished manuscript.