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

TL;DR

This paper explores the Schur product of evaluation codes, introduces new quantum codes with improved parameters, and develops PIR schemes that outperform existing methods, by leveraging algebraic structures like affine variety codes.

Contribution

It generalizes the Schur product to J-affine variety codes, constructs better CSS-T quantum codes, and proposes superior PIR schemes based on hyperbolic and affine variety codes.

Findings

Constructed quantum codes with improved parameters.

Developed PIR schemes outperforming existing ones.

Extended Schur product analysis to J-affine variety codes.

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.