In this work, we study the componentwise (Schur) product of monomial-Cartesian codes by exploiting its correspondence with the Minkowski sum of their defining exponent sets. We show that $ J$-affine variety codes are well suited for such products, generalizing earlier results for cyclic, Reed-Muller, hyperbolic, and toric codes. Using this correspondence, we construct CSS-T quantum codes from weighted Reed-Muller codes and from binary subfield-subcodes of $ J$-affine variety codes, leading to codes with better parameters than previously known. Finally, we present Private Information Retrieval (PIR) constructions for multiple colluding servers based on hyperbolic codes and subfield-subcodes of $ J$-affine variety codes, and show that they outperform existing PIR schemes.

翻译：本文通过利用单项式-笛卡尔码与其定义指数集的Minkowski和之间的对应关系，研究了这类码的分量（Schur）积。我们证明了$J$-仿射簇码特别适用于此类乘积，从而推广了先前关于循环码、Reed-Muller码、双曲码和环面码的结果。利用这一对应关系，我们分别从加权Reed-Muller码和$J$-仿射簇码的二进制子域子码构造出CSS-T量子码，所得码的参数优于已知结果。最后，我们基于双曲码和$J$-仿射簇码的子域子码，提出了面向多服务器共谋场景的私有信息检索（PIR）构造方案，并证明其性能优于现有PIR方案。