The Study On The Construction Of Some Classes Of MDS Codes

MDS codes are extremely important codes in the field of algebraic coding,which have good error correction ability.The construction of MDS codes is an important problem in the field of algebraic coding,which has been concerned and studied for a long time.Generalized Reed-Solomon codes,or GRS codes for short,are also one of the important members in the field of algebraic coding.The extended GRS codes are obtained by adding the infinite elements to GRS codes.They are good tools on studying the construction and application of MDS codes.In this dissertation,the construction of MDS Euclidean self-orthogonal codes(including self-dual codes),Euclidean hull of MDS codes and the construction of quantum MDS codes are studied by using(extended)GRS codes as tools.The specific contents are as follows:In Chapter 3,criterions of MDS Euclidean self-orthogonal codes are presented,which is a pioneering work.New MDS Euclidean self-orthogonal and self-dual codes are constructed via the criterions.In particular,among our constructions,for large square q,about 1/8·q new MDS Euclidean self-dual codes over Fq can be produced,which is much more than the total of previous results.Besides,we can construct about 1/4·q new MDS Euclidean self-orthogonal codes with different even lengths n and dimension n/2-1.In Chapter 4,we propose a mechanism for the constructions of MDS codes with arbitrary dimensions of Euclidean hulls.Precisely,we construct(extended)GRS codes with assigned dimensions of Euclidean hulls from self-orthogonal(extended)GRS codes.It turns out our constructions are more general than previous works on Euclidean hulls of(extended)GRS codes.In Chapter 5,we present three new classes of q-ary quantum MDS codes by uti-lizing GRS codes over Fq2,which satisfy Hermitian self-orthogonal property.Among our constructions,the minimum distance of some q-ary quantum MDS codes can be bigger than q/2+1.Comparing to previous known constructions,the lengths of codes in our constructions are more flexible.Moreover,our method of construction is innovative.