New Results on GDDs, Covering, Packing and Directable Designs with Block Size 5

作者：Abel R Julian R^*; Assaf Ahmed M; Bluskov Iliya; Greig Malcolm; Shalaby Nabil

来源：Journal of Combinatorial Designs, 2010, 18(5): 337-368.

DOI：10.1002/jcd.20253

• packing design
• covering design
• group divisible design
• GDD
• directed packing design
• Schonheim bound

This article looks at (5, lambda) GDDs and (v, 5, lambda) pair packing and pair covering designs. For packing designs, we solve the (4t, 5, 3) class with two possible exceptions, solve 16 open cases with lambda odd, and improve the maximum number of blocks in some (v, 5, lambda) packings when v small (here, the Schonheim bound is not always attainable). When A=1, we construct v = 432 and improve the spectrum for v = 14, 18 (mod20). We also extend one of Hanani's conditions under which the Schonheim bound cannot be achieved (this extension affects (20t + 9, 5, 1), (20t + 17, 5, 1) and (20t + 13, 5, 3)) packings. For covering designs we find the covering numbers C(280, 5, 1), C(44, 5, 17) and C(44, 5, lambda) with lambda 13 (mod 20). We also know that the covering number, C(v, 5, 2), exceeds the Schiinheim bound by 1 for v=9, 13 and 15. For GDDs of type g(n), we have one new design of type 30(9) when lambda = 1, and three new designs for lambda = 2, namely, types g(15) with g is an element of {13, 17, 19}. If lambda is even and a (5, lambda) GDD of type g(n) is known, then we also have a directable (5, lambda) GDD of type g(n).

• 出版日期2010-9