Constructions Of Special Type Of MDS Codes Over Finite Fields

Maximum distance separable(MDS)codes have good properties and are important in coding theory,due to their theoretical significance and practical applications in quantum mechanics,distributed storage systems,error-correction codes,representation of matroids,threshold sharing schemes and etc.As a class of best-known family of MDS codes,generalized Reed-Solomon codes(GRS for short)are an important class of error-correcting codes.The extended generalized Reed-Solomon codes(EGRS for short)are obtained by adding one coordinate(an infinite element)to GRS codes.Self-dual codes are important classes of linear codes that have been extensively studied.They have perfect algebraic structures and are related to polynomials and lattices.Some well-known linear codes are self-dual codes.A linear code with a complementary dual(or an LCD code)is a code which satisfies that the intersection of the code and its dual is trivial.It is shown that asymptotically good LCD codes exist.Moreover,LCD codes are shown to provide an optimum linear coding solution for the two-user binary adder channel.Therefore,the research on the intersection of self-dual codes,MDS codes and LCD codes over finite fields is an interesting research topic.This thesis devotes to the study of MDS self-dual codes and LCD MDS codes over finite fields.We first study the existence and construction of MDS Euclidean self-dual codes.The main tool is using(Extended)GRS codes.Then,we use the h-Galois inner product,which is a generalization of Euclidean inner product and Hermitian inner product,to study h-Galois LCD MDS codes.The main results of the thesis are as follows.In Chapter 3,we first provide some lemmas,which are used in the proofs of theorems.Then we construct several classes of MDS Euclidean self-dual codes with new parameters over finite fields of odd characteristic by using(Extended)GRS codes.Precisely,by choosing suitable subsets of Fq and under certain conditions,we construct q-ary MDS Euclidean self-dual codes of lengths n=tm,n=tm+1,n=tm+2 and n=p2e+1 for square q.Besides,we also produce q-ary MDS Euclidean self-dual codes of lengths n=2tpe,n=trz+1,n=(t+1)rz and n=pe+1 for prime power q with p being an odd prime.In Chapter 4,we use the h-Galois inner product to study h-Galois LCD MDS codes.We first study the structure of h-Galois dual codes of GRS and EGRS codes,and then we propose a mechanism to construct h-Galois LCD MDS codes via(Extended)GRS codes over finite fields(Lemma 4.5 and Lemma 4.7).Moreover,we construct several new families of h-Galois LCD MDS codes through GRS and EGRS codes of lengths n=rl,n=rl+1,n=pl,n=pl+1,n=rh and n=rh+1 for prime power q with p being an odd prime under certain conditions.