| Large set of designs is an important topic of design theory,which has some valuable applications to experimental design,coding theory,threshold schemes and so on.The earliest large set of designs problem called large sets of Kirkman triple systems(LKTS)has not been completely resolved for more than 160 years because of its difficulty.The common large set of designs completely resolved mainly contain: large sets of Steiner triple systems(LSTS)、 large sets of Mendelsohn triple systems(LMTS)、 large sets of Directed triple systems(LDTS)、 large sets of group divisible design(LGDD)and so on.Though the topic has acquired numerous great advances in recent years,due to its difficulty,the progress is not more than expected.Large sets of group divisible design(LGDD)as a class of important large set of designs attract much attention.This kind of large set of designs is a natural generalization of the large sets of Steiner triple systems.But the large sets of group divisible design(LGDD)were first studied because of their connection with perfect threshold schemes.In 1997 years,the spectrum of LGDD(m~v)had been completely resolved by professor Lei Jianguo.As the natural generalization of LGDD(m~v),we mainly talk about the existence of two oriented large sets of GDD design LHMTS(m~v)and LHDTS(m~v)in this article.In this article,we establish the existence of an LHMTS(m~v)for ≡ 2(mod 6)andm~v≡ 3(mod 6).Through directed and recursive construction,we can get that there exists an LHMTS(m~v)if and only if (-1)m~2≡ 0(mod 3)except possibly for = 6,m~v≡ 1,5(mod 6)and m~v?= 1.By the similar way,the existence of LHDTS(m~v)is completely determined,i.e.,there exists an LHDTS(m~v)if and only if (-1)m~2≡ 0(mod 3). |
