Research On Construction Of Binary Subspace Codes With Packet Length 8 By The EA Method

Subspace codes is a particular class of error-correcting codes with underlying alphabet the set of subspaces of a projective geometry over the finite field.The subspace codes is different from the traditional error-correcting codes.Every subspace codeword is a subspace and subspace distance depends on the performance of the codewords and the capacity in error-correcting.When all the codewords of subspace codes have same dimension,it is Constant Dimension Code which will be discussed in this thesis.The representing method of constant dimension code is(n,M,d;k)q code,in which the dimension of all the codewords is k,all the codewords are from a n-dimension projective space based on Fq,the minimum subspace distance of any two codewords is d,and the number of codewords is M.If the four parameters of constant dimension code:n,d,k,q are confirmed,the maximum number of code words of the constant dimension code:Aq(n,d;k)-its upper bound and lower bound have always been the focus of academic research,they are also the focus of this thesis.In the study of traditional error-correcting codes,we always use algebraic coding theory to study.Similarly,this thesis will use algebraic coding theory to study the upper bound of constant dimen-sion codes.In addition,this thesis will try more efficient coding method for constant dimension codes based on LMRD code(Lifted Maximum Rank Distance Codes),and present some thorough discussion on it.All codewords of subspace coding will be discussed and analyzed in projective space and vector space.There is something same and something different between these two space.However,it is impossible for the analysis of constant dimension codes without one of them.The first research point of this thesis is to offer a new solution to transform the problem of the upper bound of subspace code into a linear optimization problem.The upper bound of subspace code can be transformed to a linear optimization problem.Further we sum up how to transform the problem of the upper bound of(v,M,d:k)q code into a linear optimization problem in a general situation and sum up the general expression.The research point is of great significance to the later research.The second research point of this thesis is the research of Partial spread based on expurgation-augmentation algorithm.LMRD code(Lifted Maximum Rank Distance codes)is a very com-mon kind of constant dimension code.We can get it from maximum rank distance code with an operation of "lifting".We can get more code words if we use the encoding method of expurgation-augmentation algorithm based on LMRD code,and also can improve the distribution structure of code word in space.In previous study,we can already get(6,77,4;3)2 and(7,329,4;3)2-two sets of optimal code words by this way.A partial k-spread of Fqn is a subset C(?)Lq(k.n)such that U ? = 0 for any U,V ? C with U? V.According to the definition,a partial k-spread of Fn has two elements at least:(1)q-ary,length n,dimension k subspace code;(2)the minimum distance 2k.We call such a code a partial spread code.Partial spread code is studied in more detail by expurgation-augmentation algorithm in this thesis.And this algorithm is implemented in theory and in the simulation.The third research point of this thesis is the applications of semifields to partial spread codes.Semifield,called non-commutative divisors,is a kind of algebraic structure.It has always been an important research item in algebra theory and in field theory.The main application of semifields is that they coordinate certain projective planes,so-called semifield planes,in the same way as Fq3 does for PG(2,q),a result due to A.Albert.In the last chapter,we also give some origination of further study and guess about expurgation-augmentation algorithm.This will be the focus of future research direction and work.