The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918376714010624 |
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| author | Bodur, Şeyma Hernando, Fernando Martínez-Moro, Edgar Ruano, Diego |
| author_facet | Bodur, Şeyma Hernando, Fernando Martínez-Moro, Edgar Ruano, Diego |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10068 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval Bodur, Şeyma Hernando, Fernando Martínez-Moro, Edgar Ruano, Diego Information Theory 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. |
| title | The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval |
| topic | Information Theory |
| url | https://arxiv.org/abs/2505.10068 |