Constructions Of Six Classes Of Entanglement-assisted Quantum MDS Codes

Entanglement-assisted quantum MDS codes are a very important type of error-correcting codes that people have paid close attention to recently.It makes fullly used of the preexisting entanglement between the sender and the receiver to improve infor-mation transmission rate.Meanwhile,they can be better to overcome quantum coherence and protect the reliability and integrity of quantum information during the transmission of information.It is an important task to study entanglement-assisted quantum MDS codes in the quantum information and quantum communication.The parameters of the entanglement-assisted quantum MDS codes constructed from constacyclic codes,compared with from classic codes,are more general.In this dissertation,the number c=|Z?(-qZ)| of entangled states,denoted by c,is obtained by decomposing the defining set of constacyclic codes.On the base of it,six classes of entanglement-assisted quantum MDS codes are constructed.The minimum distances of the first five classes of entanglement-assisted quantum MDS codes are all greater than q/2+1,and so these codes have good error-correcting capabilities.In addtion,we construct five families of entanglement-assisted quantum error-correcting codes with maximal entanglement.At the end,we find out a new method different from Mehmet E.Koroglu to calculate the number of entangled states c.The number of elements in the set Z?(-qZ)is the same as the number of solutions of the congruence equation.It becomes not only the calculation process more simple,but also the defining sets more general for codes with the same length n.A great number of entanglement-assisted quantum MDS codes can be so that constructed.The performance of the entanglement-assisted quantum MDS codes is mainly evaluated by the minimum distance d and the net rate(?).To construct entanglement-assisted quantum MDS codes with constacyclic codes.Under the deter-mined length n,we already know the relationship between the minimum distance d and the defining set size ? and the relationship between the number of information bits k and the defining set size ?.If we can know the relationship between the number of en-tangled states c and the defining set size ?,we can judge the quality of the code is good or bad.Therefore,this article clarifies the piecewise function relationship between c and?,which is undoubtedly meaningful.