Construction Of Some New Quantum MDS Codes

Quantum error correction codes play an important role in quantum information pro-cessing and quantum computing.Comparing with the known classical error correction code technology,quantum error correction code technology can greatly improve the secu-rity of transmission information and the capacity and efficiency of transmission channels.Q-nary quantum MDS codes have better error correction ability and practicability,and become one of the most important class of quantum error correction codes.Therefore,constructing quantum MDS codes are important in theory and application.In recent years,many different types of quantum MDS codes are constructed,but except for a few,the minimum distances of almost all q-ary quantum MDS codes are less than or equal to q/2+1.Based on the generalized Reed-Solomon codes,the Constacyclic codes and the Hermitian construction methods,five new types of quantum MDS codes are construct-ed in this dissertation.Their minimum distances are greater than q/2+1 under certain conditions.The four new kinds are obtained by Hermitian construction methods and the gener-alized Reed-Solomon codes.Zhang et al's in 2017 constructed the q-ary quantum MDS codes with q=2am-1.The first and third kinds extend Zhang et al's work to q=hm-1.These contain the cases not only the odd prime power,but also the even prime power.The code length n=bm(q+1)is then more general.The second kind of codes,on the base of the first kind,is constructed by additional condition(h+b,2)=1.They have a larger minimum distance and so their error correction ability are better.Shi et al in 2017 considered the case q=hm+1.On the base of it,the fourth kind quantum MDS codes with larger minimum distances is obtain by means of different ways from Shi et al.In the most cases,the minimum distances of the four new quantum MDS codes are greater than q/2+1.With Hermitian self-orthogonality,the fifth kind of quantum MDS codes with length n=(q2+1)/a are constructed by the Constacyclic codes,which extend Zhang et al's work in 2015 and Lu et al' work in 2018.When a=1 or a=5,the minimum distance are greater than q/2+1.The research results obtained in this dissertation are generalization and help to study forward the quantum MDS codes in the later.