Two Constructions Of MDS Codes

Let C be an[n,k,d]linear code over a finite field Fq.When the parameters of C satisfy:d=n-k+1,we call C a maximum distance separable(MDS)code.MDS codes are of great significance in practical applications.For example,MDS codes can be well used in distributed storage systems and random error channels.The most prominent MDS codes are generalized Reed-Solomon(GRS)codes and extended GRS codes.GRS codes are determined by evaluation points and evaluation polynomials.In 2017,twisted Reed-Solomon(TRS)codes were first introduced by Beelen et al.In general,TRS codes do not necessarily lead to MDS codes.Beelen et al.gave three constructions of MDS TRS codes by taking subsets of G?{0} or V?{?} as evaluation points,where G and V are proper subgroups of Fq*and Fq,respectively.Afterwards,they generalized the above constructions by adding extra monomial(twists),and obtained a construction of MDS TRS codes.In the same paper,they also showed that most of TRS codes are not GRS codes.In this paper,we study MDS TRS codes and linear complementary dual(LCD)MDS codes.In Section 3,we give two constructions of MDS TRS codes by general-izing the evaluation points given by Beelen et al.In some cases,our constructions can get MDS codes with a longer code length than the previous results.Further,we study the equivalence of the constructed MDS codes and GRS codes.In Section 4,we construct two classes of LCD MDS codes by TRS codes.