The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval
Abstract
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 -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 -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 -affine variety codes, and show that they outperform existing PIR schemes.
Keywords
Cite
@article{arxiv.2505.10068,
title = {The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval},
author = {Şeyma Bodur and Fernando Hernando and Edgar Martínez-Moro and Diego Ruano},
journal= {arXiv preprint arXiv:2505.10068},
year = {2026}
}