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