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The Asymptotic Existence of Group Divisible t ‐Designs of Large Order with Index One
Author(s) -
Mohácsy Hedvig
Publication year - 2013
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21365
Subject(s) - mathematics , integer (computer science) , group (periodic table) , combinatorics , block (permutation group theory) , order (exchange) , additive group , discrete mathematics , block size , type (biology) , transversal (combinatorics) , arithmetic , mathematical analysis , ecology , chemistry , organic chemistry , key (lock) , computer science , economics , biology , programming language , finance
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 m u 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, t ≤ k ≤ u . Then there exists an integerm 0 = m 0 ( t , k , u )such that there exists a group divisible t ‐design of group type m u with block size k and index one for any integer m ≥ m 0satisfying the necessary arithmetic conditionsm t − iu − i t − i≡ 0 modk − i t − iThe u = k 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an existence theorem for transversal t ‐designs of large order, which was previously proved by J. L. Blanchard in an unpublished manuscript.