The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval

In this work, we study the Schur (componentwise) 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.