Study On Constant Dimension Subspace Codes

Random network coding is a powerful tool for disseminating information in networks,yet it is susceptible to packet transmission errors caused by noise or international jamming.Indeed,in the most naive implementations,a single error in one received packet would typ-ically render the entire transmission useless when the erroneous packet is combined with other received packets to deduce the transmitted message.Therefore,the network coding must have some abilities for error correction.Codes in the Grassmannian gained recently lots of interest due to the work by Koetter and Kschichang,where they presented an appli-cation of such codes for error correction in random network coding.In this paper two classes of constant dimension codes are discussed.One is cyclic orbit codes and another is constant dimension codes which contain lifted maximum rank distance codes.Results presented in this paper can be summarized as follows:1.Give a method to compute the cardinalities of cyclic orbit codes;2.Prove that if the cyclic orbit code(U,σ)is a spread code,then σ has no nontrivial invariant subspaces.In particular,if σ is of order qn-1,then(U,σ)is equivalent to the spread code constructed in[11];3.An upper bound is given for the constant dimension codes(n,M,2k,k)q which con-tain the lifted MRD codes(n,k,k)q.When f | n,the optimal codes are constructed,and when k| n,a class of constant dimension codes(n,M,2k,k)q are constructed,which have the maximum number of codewords.