  With entanglement-assisted (EA) formalism, arbitrary classical linear codes
are allowed to transform into EAQECCs by using pre-shared entanglement between
the sender and the receiver. In this paper, based on classical cyclic MDS codes
by exploiting pre-shared maximally entangled states, we construct two families
of $q$-ary entanglement-assisted quantum MDS codes
$[[\frac{q^{2}+1}{a},\frac{q^{2}+1}{a}-2(d-1)+c,d;c]]$, where q is a prime
power in the form of $am+l$, and $a=(l^2+1)$ or $a=\frac{(l^2+1)}{5}$. We show
that all of $q$-ary EAQMDS have minimum distance upper limit much larger 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.
