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.

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单项式笛卡尔码及其对偶，适用于 LCD 码、量子码和本地可恢复码

单项式笛卡尔代码是通过在笛卡尔积上评估一组单项式而定义的评估代码。它是文献中一些代码族的概括，例如复曲面代码、仿射笛卡尔代码和 J 仿射变体代码。在这项工作中，我们使用笛卡尔积的消失理想来描述单项式笛卡尔码的对偶。然后我们用对偶的这种描述来证明量子纠错码和MDS量子纠错码的存在。最后，我们证明了单项式笛卡尔码的直接乘积是具有 t 可用性的局部可恢复码，如果至少 t 个分量是局部可恢复码。