Combinatorial And Algebraic Constructions For Subspace Codes

In 2000,Ahlswede et al.proposed a new network data transmission method,named by network coding.Compared with traditional routing networks,network coding improves the multicast throughput and the security of network data transmission,and reduces the consumption of network bandwidth resources and the transmission energy consumption of nodes,which has been highly concerned by scholars all over the world.Because the channel of wireless network is usually time-varying,it is easy to cause packet loss and error.To solve this problem,K ¨otter and Kschischang in 2008 proposed the transmission model of subspace codes.Constant dimension codes(CDCs),as an important and special subspace codes,have been paid more attention.In this thesis,we are interested in constructions of subspace codes and related rank-metric codes.This thesis is organized as follows.Chapter 1 gives a brief introduction to the background of subspace codes.Chapter 2 presents constructions of constant dimension codes which contain a lifted MRD code.? A family of new codes,named rank-metric codes with given ranks(GRMCs),is introduced to generalize the parallel construction and to give a lower bound for all CDCs.Further,a Singleton-like upper bound and a lower bound for GRMCs are established by using Gabidulin codes.? Two new constructions are established by combining parallel construction and multilevel construction.The first construction shows the sufficient condition if the two constructions can be combined.Applying the construction,the lower bound of(n,2?,k)-CDCs is given,where n ? 2k + ?,k ? 2?.Further,the lower bounds of(n,4,k)q-CDCs and(n,4,5)q-CDCs are also improved.The second construction shows that the sufficient condition is not a necessary condition.Applying the construction,the lower bound of(4?,2?,2?)q-CDCs is improved.Further,we calculate its ratio between the lower bound and the upper bound.The ratio is greater than 0.967688 for ? ? 2 and any prime power q,and greater than 0.999260 for? ? 3 and any prime power q.In Appendix D,many CDCs with larger size than the previously best known codes at the online tables http://subspacecodes.unibayreuth.de are given.Chapters 3 and 4 show constructions of optimal Ferrers diagram rank-metric codes(FDRMCs).? By using generator matrices of the systematic MRD codes,two classes of new optimal FDRMCs are given.By introducing restricted Gabidulin codes,we obtain another class of optimal FDRMCs,which generalizes all constructions by subcodes of Gabidulin codes.? By introducing the concept of proper combinations of Ferrers diagrams,we show the necessary condition combining small FDRMCs to a bigger one,and obtain two new constructions.Applying the two constructions,the first class of optimal FDRMCs whose optimal dimensions is not equal to v0 for any prime power is given.Further,the optimality of FDRMCs with rank distance 3 and unknown minimum size of Ferrers diagram is solved.? We provide two constructions for FDRMCs based on two different ways to represent elements of a finite field Fqm(vector representation and matrix representation).Also,the constructions can be generalized to deal with non-linear FDRMCs.? Each of these constructions produces new optimal codes with different diagrams and parameters.In Chapter 5,we summarize this dissertation and give several problems for further research.