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 • Information Theory
Information Theory
AI summaryⓘ
The authors study special types of codes called cyclic codes over a particular mathematical structure made from two binary fields combined. They find specific polynomials that describe parts of these codes related to their Hermitian properties. Using these results, they build quantum error-correcting codes through a method called Quantum Construction X. Additionally, they create entanglement-assisted quantum codes by applying techniques involving matrix product codes on related codes with complementary dual properties over the same structure.
Cyclic CodesHermitian HullQuantum Error-Correcting CodesQuantum Construction XEntanglement-Assisted Quantum CodesMatrix Product CodesLinear Complementary Dual (LCD) CodesComposite RingsFinite FieldsCode Polynomials
Authors
Rabia Zengin, Mehmet Emin Köroğlu
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.