Generalized Doubly Resolvable Packing And The Corresponding Codes

Constant composition code(CCC) is a generalization of binary constant weight code.The constant composition codes have arisen recent interest due to their numerous applications in frequency hopping sequences,powerline communications and so on.Ding etc.proved that an optimal CCC is equal to an optimal generalized doubly resolvable packing(GDRP).This dessertation investigates the structure,the construction and the existence of optimal GDRPs.Making use of GDRPs,we establish a series of combinatorial methods of constructing optimal CCCs and some new families of optimal CCCs.In 2003,Luo etc.established an upper bound of CCC(an LFVC bound in short), which has become a benchmark of many researchers measuring the optimality of CCC. Chapter 2 first gives a combinatorial proof of the LFVC bound,then imposes the structure of optimal GDRPs meeting the LFVC bound,finally builds some new upper bounds of GDRPs,which are our benchmarks to measure the optimality of GDRPs.Chapter 3 discusses the construction and the existence of optimal GDRPs meeting the LFVC bound.Firstly,we introduce several new auxiliarly designs and present some valid methods of constructing optimal GDRPs.Secondly,we completely solve the existence problem of optimal GDRPs satisfying 3λ=2μ,almost solve the existence problem of optimal GDRPs satisfying 4λ=3μwith only two possible exceptions, and obtain an asymptotic result for optimal GDRPs satisfying 5λ=4μ,whereλandμare any positive integers.The corresponding optimal CCCs are then obtained. Meanwhile,we improve the existence results of complementary frames and NGBTDs given by Lamken.In Chapter 4,we discuss the structure and the construction of optimal GDRPs meeting the generalized LFVC bound.A difference family with the prescribed properties, called resolvable DF,is introduced and used to establish constructions and existence results of optimal GDRPs.Some other constructions of optimal GDRPs are provided in Chapter 5.In the final chapter,we present some open problems for further study.