The Existence And Application Of A Class Of Generalized Packing Designs

In 2013,R.F.Bailey and A.C.Burgess proposed the concept of generalized packing design under a unified perspective.It is an extension of packing design and generalized t-design,and is closely related with Howell design,(partial)Latin square,(resolvable)triple system,graph colouring etc.,which can be used to construct(multiple)constant weight codes.Generalized packing design not only has an important position in combinatorial design theory,but also has a wide range of applications in coding field.There are few known results on generalized packing designs,which are mainly con-centrated on the cases of t=2,k=3,4.In this paper,we will study the construction,existence and application of the optimal generalized packing design with t=2,k=5 and block type k=(2,1,1,1).This paper is divided into four parts.The first part introduces some basic concepts and known results,and gives the relevant definitions of multiple constant weight codes.The second part,based on some auxiliary designs such as d-dimensional Howell design,gives two recursive construction methods including PBD construction method and frame construction method.Furthermore,we study the existence of H3(s,2n)for n<s<2n-1.The third part gives some optimal results of the generalized 2-(v,k,1)packing design with k=(2,1,1,1).In the fourth part,corresponding to the relationship between the generalized packing design and multiple constant weight codes,some results of the optimal multiple constant weight codes with the minimum distance d=8 and weight w=(2,1,1,1)are obtained.