Monotonic Directed Designs

Difference triangle sets have many interesting applications in data communications. Perfect difference family is an optimal difference triangle set.Ge,Ling and Miao construct radar arrays with difference family[8].It's easy to know difference triangle set,radar array and difference family are highly related to each other in design theory.The notion of monotonic directed designs(MDDs) was first introduced by Chu and Colbourn in[4]for the construction of difference triangle sets.Although the existence of directed designs has been extensively studied,few families of MDDs,are known currently.However,we should not expect any MDD with block size greater than 6.An example of MDD,not constructed from a regular difference triangle set, would be of great interest.The main result are as follows:1.For the case of k = 3,the necessary condition for the existence of MDD(v,3) is v≡0,1,4,9(rood 12),v≥4.We need to prove the necessary condition is also sufficient.First,we can derive MDDs from difference family when v≡1,4(mod 12).In order to find the remaining designs,we need find some small designs which are necessary for the construction.With some GDDs that already exist,we can find the remaining MDDs.So,we can come to the conclusion:an MDD(v,3) exists if and only if v≡0,1,4,9(mod 12).2.For the case of k≡4,the necessary condition for the existence of MDD(v,4) is v≡1(mod 3).When v≡1(mod 6),MDDs can be derived from difference family,which is still unsettled.When v≡4(mod 6),we give the proof for the nonexistence of MDD(v,4),v = 4,10.We also found some small MDD(v,4) when v≡4(mod 6),but it's not sufficient enough for the recursive construction. The above are the main results of this thesis.In order to construct difference triangle sets,we need more MDDs with block size 4 and 5.Of course,it would be much more difficult to find them.We need much more efficient algorithm in searching. That needs further study and exploration.