Constructions Of Maximum Distance Separable Codes With Specific Duality Properties

Maximum Distance Separable(MDS)codes are of the largest minimum Hamming distance for a fixed code rate and code length.As the minimum Hamming distance of an error-correcting code represents its ability to correct or detect errors,MDS codes are optimal in this sense.In practice,MDS codes have witnessed successful applications in storage systems and communication systems.A linear code has the algebraic structure of a vector space.Linear codes are central to the theory of error-correcting codes in the sense that they share nice algebraic structures,simple representations and efficient encoding and decoding algorithms.A lineal code C always comes with its dual code C⊥.If C and C⊥ further satisfy certain conditions,C is said to be a linear code of specific duality properties.In particular,the Self-Dual property,which requires C=C⊥,and the Linear Complementary Dual(LCD)property which requires C n C⊥={0},will be concerned in this thesis.Linear codes of specific duality properties are important in the sense that they bridge the theory of error-correcting codes,cryptography and the theory of quantum error-correcting codes.The MDS property and certain duality properties are both important to the theory of error-correcting codes,and it is natural to consider the construction of error-correcting codes sharing both MDS and duality properties.On the other hand,MDS codes with specific duality properties have found successful applications in related area.Taking Self-Dual MDS codes and LCD MDS codes which will be mainly concerned in this thesis as examples:Self-Dual codes give rise to the construction of additive quantum codes,while MDS Self-Dual codes yield additive quantum codes satisfying the quantum MDS property.LCD codes can be used to resist side-channel attacks,while its ability to resist such attacks is related to its minimum distance.In conclusion,it is interesting to further investigate the construction of MDS codes with specific duality properties.This thesis investigates the construction of MDS codes with specific duality properties,and mainly consists of the following parts:(1)Constructions of MDS Self-Dual codes:There are a lot of related works in the literature,producing MDS Self-Dual codes of different parameters.However,these results failed to cover all possible parameters of MDS Self-Dual codes.This thesis seeks to propose some constructions of MDS Self-Dual codes of new parameters.In particular,for any odd prime power r and any even length n∈[2r,3r],q=r2-ary MDS Self-Dual codes of length n are proposed in this thesis.This construction of MDS Self-Dual codes of consecutive lengths extends the already known constructions producing MDS Self-Dual codes of consecutive lengths in the range[1,2r].On the other hand,for each singly even length n∈[3r,4r],q-ary MDS Self-Dual codes of length n are constructed explicitly.Furthermore,some generalized constructions are also proposed.The code length of these constructions no longer ranges over a consecutive interval,and it is somewhat difficult to actually determine the code length as they depend on the factorization of q-1.Experiments and statistics show that all constructions in this thesis contribute to around 0.15q/2～0.2q/2 different new lengths.(2)Constructions of MDS LCD codes:Similar to the construction of MDS SelfDual codes,there are a lot of related works concerning the constructions of MDS LCD codes in the literature.Yet,only a portion of possible parameters of MDS LCD codes has been covered by these results.It is well-known that the Generalized Reed-Solomon(GRS)codes are MDS codes.In this thesis,an equivalent characterization of whether a GRS code satisfies the LCD condition or not,is proposed.By this equivalent characterization,a concatenation construction and a Vandermonde-matrix-based construction of LCD GRS codes are proposed.These new methods takes Self-Dual GRS codes as inputs and outputs LCD GRS codes,which bridges the construction of MDS LCD codes and the construction of MDS Self-Dual codes.Some constructions of MDS LCD codes of new parameters are discovered through these methods.Experiments and statistics show that for a fixed odd prime power r,the number of new q=r2-ary MDS LCD codes proposed in this thesis is around 0.4q2/4.(3)Constructions of Hermitian LCD MDS codes:The LCD property discussed above is given by the Euclidean dual code.We can also consider the Hermitian LCD property given by the Hermitian dual code.As there are connections between the Euclidean dual code and the Hermitian dual code of a linear code,this thesis seeks to generalize the results in(2)to the case of Hermitian LCD codes.In particular,an equivalent characterization of whether GRS codes satisfy the Hermitian LCD property or not,is proposed,yielding a concatenation construction for Hermitian LCD GRS codes.These results can be though of as the generalization of the corresponding results in(2).The innovation of this thesis lies at two points.On one hand,MDS Self-Dual codes,MDS LCD codes and MDS Hermitian LCD codes of new parameters are constructed explicitly,and statistics shows that these new parameters take up a proportion strictly greater than zero among all possible parameters.On the other hand,new methods for the construction of LCD GRS codes are proposed.Furthermore,these new methods relate the construction of Self-Dual GRS codes and LCD GRS codes so that results on these two problems can be communicated.