QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over $\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)$

2026-06-01

AI Summary

This study investigates the construction of high-performance quantum error-correcting (QEC) codes and entanglement-assisted quantum error-correcting (EAQEC) codes from cyclic codes over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$. By analyzing for the first time the Hermitian hull and Hermitian sum structures of cyclic codes over this ring, the work integrates quantum Construction X, matrix-product codes, and linear complementary dual (LCD) code techniques to propose novel constructions of QEC and EAQEC codes. This approach not only extends the design framework for EAQEC codes but also yields several new families of quantum codes, thereby enriching the theoretical foundation of quantum error correction based on non-traditional algebraic structures.

Abstract

In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$, where $v^2 = v$. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.

Problem

• Quantum Error-Correcting Codes
• Entanglement-Assisted Quantum Error-Correcting Codes
• Hermitian Hulls
• Cyclic Codes
• Composite Rings

Innovation

• Hermitian hull
• Quantum Construction X
• Matrix product codes
• Entanglement-assisted quantum error-correcting codes
• Cyclic codes over rings