Entanglement-assisted Quantum MDS Codes From Constacyclic Codes

Entanglement-assisted quantum error-correcting codes(EAQECCs)are used to transmit information by the pre-existing shared entanglement between the sender and the receiver,thus improving the transmission rate.Based on entanglement-assisted stabilization,we can construct entanglement-assisted quantum error-correcting codes from any classical linear codes.And it is a relatively simple method to construct them by the decomposition of the defining set of constant cyclic codes.The number of entangled bits is also determined by the decomposition of the defining set of constant cyclic codes.In this paper,based on previous works,we construct two classes of entanglement-assisted quantum MDS codes with different parameters.The main results are as follows:1.Entanglement-assisted quantum MDS code with length n=8(q-1).Let q=8t-1 be an odd prime power,where t is an even positive integer and t≥4,n=8(q-1).(1)When 1≤j≤8,EAQMDS codes with parameters[[8(q-1),8(q-1)+22(4t+6+j),4t+7+j;2]]q are exist;(2)When 1≤j≤15,EAQMDS codes with parameters[[8(q-1),8(q-1)+42(4t+6+j),4t+7+j;4]]q are exist.2.Entanglement-assisted quantum MDS code with length n=q2+1/13.Let q=26m+5 be an odd prime power,where m is a positive integer.When 1≤j≤6m+1,EAQMDS codes with parameters[[q2+1/3,q2+1/13+4-2(2j+1),2j+4-2(2j+1),2j+1;4]]q and[[q2+1/13,q2+1/13-20m-4j+4,10m+2j+3;8]]q are exist.