  Quantum error correcting codes play the role of suppressing noise and
decoherence in quantum systems by introducing redundancy. Some strategies can
be used to improve the parameters of these codes. For example, entanglement can
provide a way for quantum error correcting codes to achieve higher rates than
the one obtained via the traditional stabilizer formalism. Such codes are
called entanglement-assisted quantum (QUENTA) codes. In this paper, we use
algebraic geometry codes to construct several families of QUENTA codes via the
Euclidean and the Hermitian construction. Two of the families created have
maximal entanglement and have quantum Singleton defect equal to zero or one.
Comparing the other families with the codes with the respective quantum
Gilbert-Varshamov bound, we show that our codes have a rate that surpasses that
bound. At the end, asymptotically good towers of linear complementary dual
codes are used to obtain asymptotically good families of maximal entanglement
QUENTA codes. Furthermore, a simple comparison with the quantum
Gilbert-Varshamov bound demonstrates that using our construction it is possible
to create an asymptotically family of QUENTA codes that exceeds this bound.
