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<p>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 <i>J</i>-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 <i>J</i>-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 <i>J</i>-affine variety codes, and show that they outperform existing PIR schemes.</p>

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

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年3月10日

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  • Şeyma Bodur,
  • Fernando Hernando,
  • Edgar Martínez-Moro,
  • Diego Ruano

摘要

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.