  The entanglement-assisted (EA) formalism allows arbitrary classical linear
codes to transform into entanglement-assisted quantum error correcting codes
(EAQECCs) by using pre-shared entanglement between the sender and the receiver.
In this work, we propose a decomposition of the defining set of constacyclic
codes. Using this method, we construct four classes of $q$-ary
entanglement-assisted quantum MDS (EAQMDS) codes based on classical
constacyclic MDS codes by exploiting less pre-shared maximally entangled
states. We show that a class of $q$-ary EAQMDS have minimum distance upper
limit greater than $3q-1$. Some of them have much larger minimum distance than
the known quantum MDS (QMDS) codes of the same length. Most of these $q$-ary
EAQMDS codes are new in the sense that their parameters are not covered by the
codes available in the literature.
