Research On Constructions Of Constant- Dimension Codes Based On LMRD Codes

Subspace code is a particular class of error-correcting code with underlying al-phabet the set of subspaces of a projective geometry over the finite field. Subspace codes are different from the traditional error-correcting codes in that the subspace distance is a metric for the capacity in error-checking and error-correcting. If all codewords have same dimension, the subspace codes would be reduced to constant-dimension codes, we call the (n, M, d; k)q code as a constant-dimension code with packet length n, constant dimension k and minimum subspace distance d, every subspace of (n, M, d; k)q code based on finite field Fq, and the number of codewords being M.The so-called main problem of subspace coding asks for the determination of subspace codes of maximum size Aq(n, d; k) when the remaining parameters are fixed. In the thesis, the main problem of (n, M, 4; 3)2 code is investigated by algebraic theory which is similar to ordinary algebraic theory for traditional error-correcting codes. All codewords of subspace coding will be analyzed in projective spaces and vector space, these two space are both useful to our thesis.In fact, the subspace coding problem is equivalent to a combinatorial opti-mization problem. This motivates us to solve the problem of (n, M,4; 3)2 code by using the especial relationship between projective plane and projective line, since this relationship causes a circulant structure which can be used in the combinato-rial optimization problem. So the first point of the thesis is the analysis of upper bound by the circulant structure, where we obtain tight upper bounds and deep understanding of subspaces of projective space, where the latter is useful to later study.LMRD code is a systematic constant-dimension code and generated by lifting maximum rank distance code, which is a well-known rank-metric code. As show in previous paper, (6,77,4; 3)2 code and (7,329,4; 3)2 code can be constructed by first expurgating and then augmenting the corresponding LMRD code in a computer-free way, these two codes are both best-known plane subspace codes. In the second point of this thesis we develop the algebraic theory of subspaces good to removal from the algebraic analysis of LMRD code, a good removable-subset of LMRD code can be rearranged into a larger set of new planes, then we also give an intuitive method to judging the subspace distance within new planes. In the thesis we generalize the expurgation-augmentation approach to arbitrary packet length n and provide a detailed theoretical analysis of our method. In the third point of the thesis we propose collision subspace and collision matrix which are used in optimization problem of new planes to obtain the optimal solution of net-gain (relative to an LMRD code). Computational results for small parameters are listed at the back of the thesis, as it turns out, our method is capable of producing codes larger than those obtained by the LMRD bound. In the last chapter, we also give some open problems related to our work, which is worthy to be studied further.