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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.

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters. Namely, we are able to provide new binary records and non-binary codes improving the ones available in the literature.

We study monomial-Cartesian codes (MCCs) which can be regarded as $(r,\delta)$-locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to $(r,\delta)$-optimal LRCs for that distance, which are in fact $(r,\delta)$-optimal. A large subfamily of MCCs admits subfield-subcodes with the same parameters of certain optimal MCCs but over smaller supporting fields. This fact allows us to determine infinitely many sets of new $(r,\delta)$-optimal LRCs and their parameters., Comment: This is a revised version of the manuscript "Optimal $(r,\delta)$-LRCs from zero-dimensional affine variety codes and their subfield-subcodes". We have modified the title and the new one is "Optimal $(r,\delta)$-LRCs from monomial-Cartesian codes and their subfield-subcodes". This new version contains rather changes, the main ones appear in Section 4

We characterize numerical semigroups $S$ with embedding dimension three attaining equality in the inequality $\max\Delta(S)+2\leq \operatorname{cat}(S)$, where $\Delta(S)$ denotes the Delta set of $S$ and $\operatorname{cat}(S)$ denotes the catenary degree of $S$.