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We construct new stabilizer quantum error-correcting codes from generalizedmonomial-Cartesian codes. Our construction uses an explicitly defined twistvector, 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 theminimum distance is $3$ the codes are at least Hermitian Almost MDS. For aninfinite family of parameters when $m=2$ we prove that our codes beat theGilbert-Varshamov bound. We also present many examples of our codes that arebetter than any known code in the literature.

MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes