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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

TL;DR

This work addresses the design of code-based CSS–T quantum codes and Private Information Retrieval (PIR) schemes through a unified treatment of the Schur (componentwise) product of evaluation codes. The authors develop an explicit Minkowski-sum description for the Schur product in families of $J$-affine variety codes and show that subfield-subcodes preserve these product structures, enabling binary CSS–T constructions with improved parameters. They also exploit transitivity and complete cyclotomic coset structures to build PIR schemes with adjustable privacy that outperform classical RM- and Berman-code-based approaches, including one- and multi-variable $J$-affine code constructions and hyperbolic-code-based retrieval. The results expand the design space for quantum codes and distributed private data retrieval, with practical implications for higher-rate, lower-overhead quantum error-correction and secure data access. The framework hints at extensions to secure multi-party computation and other multiplicative-secret-sharing settings by clarifying how duals and star-products interact in these code families.

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.

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

TL;DR

This work addresses the design of code-based CSS–T quantum codes and Private Information Retrieval (PIR) schemes through a unified treatment of the Schur (componentwise) product of evaluation codes. The authors develop an explicit Minkowski-sum description for the Schur product in families of -affine variety codes and show that subfield-subcodes preserve these product structures, enabling binary CSS–T constructions with improved parameters. They also exploit transitivity and complete cyclotomic coset structures to build PIR schemes with adjustable privacy that outperform classical RM- and Berman-code-based approaches, including one- and multi-variable -affine code constructions and hyperbolic-code-based retrieval. The results expand the design space for quantum codes and distributed private data retrieval, with practical implications for higher-rate, lower-overhead quantum error-correction and secure data access. The framework hints at extensions to secure multi-party computation and other multiplicative-secret-sharing settings by clarifying how duals and star-products interact in these code families.

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.
Paper Structure (14 sections, 19 theorems, 61 equations, 11 tables)

This paper contains 14 sections, 19 theorems, 61 equations, 11 tables.

Table of Contents

  1. Introduction
  2. (Affine variety) Monomial Cartesian codes and $J$-affine variety codes
  3. $J$-Affine variety codes
  4. Subfield subcodes of $J$-Affine codes
  5. CSS-T codes
  6. CSS-T codes from Weighted Reed-Muller codes
  7. CSS-T codes from subfield subcodes of $J$-affine codes
  8. Private Information Retrieval
  9. Transitivity
  10. PIR from hyperbolic codes
  11. PIR with subfield-subcodes of $J$-affine variety codes
  12. Using subfield subcodes of $J$-Affine Variety Codes in One Variable
  13. Using Hyperbolic Codes
  14. Comparison with Berman Codes

Key Result

Proposition 1

Let $J \subseteq \{ 1 , 2, \ldots , m\}$, consider $\boldsymbol{a}, \boldsymbol{b} \in \mathcal{H}_J$ and let $X^{\boldsymbol{a}}$ and $X^{\boldsymbol{b}}$ be two monomials representing elements in $\mathcal{R}_J$. Then, $\mathrm{ev}_J ( X^{\boldsymbol{a}}) \cdot \mathrm{ev}_J (X^{\boldsymbol{b}})$

Theorems & Definitions (42)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • proof
  • Theorem 4
  • Remark 5
  • Lemma 6
  • proof
  • Remark 7
  • Theorem 8
  • ...and 32 more