  A monomial-Cartesian code is an evaluation code defined by evaluating a set
of monomials over a Cartesian product. It is a generalization of some families
of codes in the literature, for instance toric codes, affine Cartesian codes
and $J$-affine variety codes. In this work we use the vanishing ideal of the
Cartesian product to give a description of the dual of a monomial-Cartesian
code. Then we use such description of the dual to prove the existence of
quantum error correcting codes and MDS quantum error correcting codes. Finally
we show that the direct product of monomial-Cartesian codes is a locally
recoverable code with $t$-availability if at least $t$ of the components are
locally recoverable codes.
