MDS, Hermitian almost MDS, and Gilbert-Varshamov quantum codes from generalized monomial-Cartesian codes.

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in m variables. When m = 1 our codes are MDS, and when m = 2 and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when m = 2 we prove that our codes beat the Gilbert-Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.