Halvings on small point sets

✍ Scribed by Reinhard Laue

• Book ID: 101296219
• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 403 KB
• Volume: 7
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(1999)7:4<233::aid-jcd1>3.0.co;2-e

✦ Synopsis

A halving is a t-design which has the same parameters as its complementary design. Together these two designs form a large set LS[2](t, k, v). There are several recursion theorems for large sets, such that a single new halving results in several new infinite families of halvings. We present new halvings with the parameters 7-(24, 10, 340), 6-(22, 9, 280), 5-(21, 10, 2184), and 5-(21, 9, 910). Recursive constructions by S. Ajoodani-Namini and G. B. Khosrovshahi [Discrete Math 135 (1994), 29-37; J. Combin. Theory A 76 (1996), 139-144] then yield that an LS[2](t, k, v) exists if and only if the parameter set is admissible for t = 6, k = 7, 8, 9, and for t ≤ 5, k ≤ 15. Thus, Hartman's conjecture is true in these cases.