Constructions of entanglement-assisted quantum MDS from generalized Reed-Solomon codes

Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing self-orthogonal condition with the help of pre-shared entanglement between the sender and the receiver. In this paper, by using generalized Reed-Solomon codes, we construct two families of entanglement-assisted quantum error-correcting MDS (EAQMDS) codes with parameters $[[\frac{b({q^2}-1)}{a}+\frac{{q^2} - 1}{a}, \frac{b({q^2}-1)}{a}+\frac{{q^2}-1}{a}-2d+c+2,d;c]]_q$, where $q$ is a prime power and $a| (q+1)$. Among our constructions, the EAQMDS codes have much larger minimum distance than the known EAQMDS codes with the same length and consume the same number of ebits. Moreover, some of the lengths of ours EAQMDS codes may not be divisors of $q^2\pm 1$, which are new and different from all the previously known ones.