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arXiv · 2505.10068

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

Şeyma Bodur·Fernando Hernando·Edgar Martínez-Moro·Diego Ruano
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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 $ 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.

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Şeyma Bodur, Fernando Hernando, Edgar Martínez-Moro, Diego Ruano. 2025-05-15. The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval. https://arxiv.org/abs/2505.10068

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