The Existence Of 2-Balanced Mendelsohn Triple Systems

A block design(X,B)is called j-balanced,when any two j-subsets of X are contained in the same or almost the same nuiber of blocks(difference at most one).The study on the balance of block designs arise in the study on the variance in file availability in distributed file system by Bermond et al.It is proved that a file storage method corresponds to a design;and a well-balanced design(j-balanced for any j less than block size)corresponds to a most reasonable file storage lethod.The sufficient and necessary conditions for the existence of well-balanced triple systems have been determined.This paper will do some research on the existence of j-balanced Mendelsohn triple systems.Based on the definition of j-balanced undirected design,this paper first proposes the concept of j-balanced Mendelsohn triple systems.Write a 2-balanced Mendelsohn triple system as 2-BMTS(v,b),write a 2-balanced and 3-balanced Mendelsohn triple system as(2,3)-BMTS(v,b).This paper mainly focuses on the existence of 2-BMTS(v,b)and(2,3)-BMTS(v,b),and finally determines their sufficient and necessary conditions.The paper is organized as follows:Firstly,we present two basic constructions and prove the necessary conditions for the existence of 2-BMTS and(2;3)-BMTS.Then combining with computer program,we solve the existence of 2-BMTS(v,b)and(2,3)-BMTS(v,b)with order v=5,6,8,14.After that we classify the order v.When v=3k or v=3k+1,we prove the spectrum of pairs(v,b)for which a 2-BMTS(v,b)and a(2,3)-BMTS(r,b)exists by employing large sets of Mendelsohn triple systems.When v=6k+5 or v=6k+2,we prove the spectrum of pairs(v,b)for which a 2-BMTS(v,b)and a(2,3)-BMTS(v,6)exists by using partitionable Mendelsohn candelabra system of order v.In conclusion,we completely determine the sufficient and necessary conditions for which a 2-BMTS(v,b)or a(2,3)-BMTS(v,b)exists.It enriches the content of combinatorial de-sign theory.This paper also involves an initial research on the existence of well-balanced Mendelsohn triple systems.It is proved that when there is a pure 2-BMTS(v,b),there may not exists a WBMTS(v,b).This lays a foundation for future research on the existence of WBMTS(v,b).