  Entanglement-assisted quantum error-correcting (EAQEC) codes make use of
preexisting entanglement between the sender and receiver to boost the rate of
transmission. It is possible to construct an EAQEC code from any classical
linear code, unlike standard quantum error-correcting codes, which can only be
constructed from dual-containing codes. However, the number $c$ of pre-shared
maximally entangled states is usually calculated by computer search. In this
paper, we first give a new formula for calculating the number $c$ of pre-shared
maximally entangled states. Then, using this formula, we construct three
classes of new entanglement-assisted quantum error-correcting
maximum-distance-separable ( EAQEC MDS) codes.
