Two Classes Of Packing Designs Based On Constant Weight Codes

Constant-weight codes(CWC)are an important class of codes in coding theory,with good error detection and correction capabilities.Constant-composition codes(CCC)and optical orthogonal codes(OOC)are special constant-weight codes.As an important class of designs in combinatorial design theory,group divisible designs(GDD)play a fundamental role in constructing other kinds of designs.Chee et al.introduced a group divisible codes(GDC)comparing with group divisible designs,and used them to construct the optimal constant-composition codes with weight three.Group divisible packings(GDP)are a class of designs closely related to group divisible designs.When constructing threedimensional optical orthogonal codes,Lidong Wang et al.introduced a special class of group divisible packings and gave them equivalence to a special class of three-dimensional optical orthogonal codes.In 2000,the theory of decompositions of complete edge-colored digraphs have been used by Lamken and Wilson to prove the asymptotic existence of many different classes of combinatorial configurations.Recently,Chee et al.constructed several classes of constant-composition codes and group divisible codes basing on this theory,and proved that Johnson bounds of the codes exist asymptotically.In this paper,we first give a method of construction group divisible codes by using decompositions of complete edge-colored digraphs.Graph family decomposition can be regarded as a kind of graph packing,except for the leave.Second,we research the construction of optimal group divisible packings problems and give the existence of some infinite classes.The main content of this article is as follows:In chapter 1,we introduce some basic concepts and known results of constant-weight codes,constant-composition codes,group divisible codes and group divisible designs and group divisible packings et al.In chapter 2,we study group divisible codes with type gn and give the definition of graph family G(w,g).Further,we establish the corresponding relationship between graph and codeword by using the decompositions of complete edge-colored digraphs,in order to obtain a method of construction group divisible codes with type gn and distance 2w-3.In chapter 3,we study the group divisible packings with type gn.Firstly,we introduce several kinds of auxiliary designs,and use them to give the recursive construction of group divisible packings.Secondly,on the basis of small parameter designs,we obtain some infinite classes of optimal group divisible packings when the number of leave edges is ng/2,ng+3/2 and ng+3.