Research On Algebraic Construction Of Constant Dimension Subspace Codes And Flag Codes

At present,information technology is changing with each passing day.The field of mobile communication has experienced more than 30 years of development from 2G to 5G and communication capabilities and performance have continued to improve.As the communication network expands and its structure becomes more and more complex,higher requirements are put forward for the reliability,effectiveness and security of information transmission and processing in modern digital communication.Network coding allows intermediate nodes in network communication to process received information,which means that in addition to simply copying and storing received information,intermediate nodes can also process it,that is,encode.This technique effectively improves the throughput of the network.However,in actual network communication,various errors are unavoidable.In order to deal with these errors effectively,the concept of network error correction is proposed.Due to the great application potential of network error correction in network communication,it has received extensive attention and developed rapidly once it was proposed.Subspace codes are a kind of network errorcorrecting codes.Constant dimension subspace codes,that is,the codewords in a subspace code have the same dimension,are the most studied type of subspace codes.A basic problem is to determine the maximum possible cardinality for constant dimension subspace codes.However,this is a difficult problem both theoretically and algorithmically.Therefore,most researches turn to improving the upper and lower bounds of the maximum possible size of constant dimension subspace codes.When considering the cyclic shift of subspaces,another important subclass of constant dimension subspace codes is obtained:cyclic constant dimension subspace codes.Cyclic constant dimension subspace codes have a good algebraic structure and efficient encoding and decoding algorithms,so they have been extensively studied by scholars at home and abroad.In addition,as a generalization of subspace codes,flag codes have a larger minimum distance and have also received extensive attention in recent years.This dissertation focuses on three aspects:the improvement of the lower bounds of the cardinality of constant dimension subspace codes,the construction of large cyclic constant dimension subspace codes and the orbital constructions of optimum distance flag codes.The main achievements and innovations of this dissertation are as follows:(1)A construction method for improving the lower bounds of constant dimension subspace codes based on the inserting construction is proposed.Taking the generator matrix of the codewords of a constant dimension subspace codes as the starting point,constant dimension subspace codes with larger cardinality than the previously best known codes are given.The specific idea is:select a generator matrix of the codewords of a special constant dimension subspace code and study its columns.By exchanging some specified columns,the generator matrices of the codewords of multiple constant dimension subspace codes are obtained.Considering the union of the corresponding constant dimension subspace codes and obtaining a class of new constant dimension subspace code.Then the constructed constant dimension subspace code can be inserted into a suitable parallel construction,so that a constant dimension subspace code with larger size is obtained.Therefore,the lower bounds of the cardinality of constant dimension subspace codes for some parameters are improved;(2)The constructions of new Sidon spaces and cyclic constant dimension subspace codes with cardinality exceeding(qn-1)/(q-1)and minimum distance remaining 2k-2.First,several new constructions of Sidon spaces are given.Secondly,by using the connection between Sidon space and cyclic constant dimension subspace codes,several classes of optimal cyclic constant dimension subspace codes are obtained and the conjecture about the existence of some parameters of optimal cyclic constant dimension subspace codes is solved.Finally,under certain conditions,considering the union of different cyclic constant dimension subspace codes,a cyclic constant dimension subspace code with larger cardinality is constructed;(3)The orbital constructions of the optimum distance flag codes based on Abelian non-cyclic groups.Firstly,a maximum rank distance code is constructed on Fq2k and an Abelian non-cyclic group is given using its codewords.An optimum distance orbit full flag code generated by a suitable flag under the action of the Abelian non-cyclic group is constructed.Secondly,an optimum distance flag codes of full admissible type vector on Fq3k is obtained through the union of multi-orbit subspace codes.Then,the idea of construction in Fq3k is generalized and an optimum distance orbit flag code of full admissible type vector on Fq(2s+1)k is given.Finally,an optimum distance flag code of type(1,…,k)on Fq2k is given through an Abelian non-cyclic group and a special lower triangular matrix.In addition,the optimum distance flag codes constructed above have reached the corresponding maximum possible cardinality.