The Construction Of Maximum Rank Distance Codes And Their Applications

Error correction coding is an important means to ensure information reliability.Rank-metric codes play an important role in error-correcting coding theory.Maximum Rank Distance?MRD?codes,as a special type of rank-metric codes,have seen much interest in recent years due to a wide range of applications including storage systems,cryptosystems,spacetime codes and random linear network coding.Delsarte and Gabidulin indepen-dently construct the first family of F_q-linear MRD codes named Gabidulin codes.Gen-eralized Gabidulin codes,twisted Gabidulin codes,generalized twisted Gabidulin codes,etc.are derived from it.Recently,Trombetti and Zhou construct a new family of MRD codes in F_q^2n×2n.In this paper,the basic theory and mathematical properties of rank-metric codes are discussed.The characterization methods of rank-metric codes based on linearized poly-nomials are given.The invariants that play an key role in determining the equivalence of rank-metric codes are studied.On this basis,the self-duality of MRD codes and the public-key cryptosystem based on linear codes are studied.By analyzing the construction methods and equivalence proofs of the new MRD codes given by Trombetti and Zhou,we obtain a judge condition when the code family is equivalent to a self-dual MRD code.By describing the vector representation of the family of Gabiduliin codes,the McEliece public-key cryptosystem based on MRD codes is given,and an improved algorithm which can resist message resend and related message attack is designed.