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.

翻译：本文中，我们确定了定义在复合环 $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$（其中 $v^2 = v$）上的循环码的Hermitian包与Hermitian和的生成多项式。基于这些结构，我们通过将量子构造X的Hermitian对偶版本应用于所获得的Hermitian包与和，构造了量子纠错（QEC）码。此外，通过将矩阵乘积码方法应用于定义在同一环上的线性互补对偶（LCD）码，我们推导出了纠缠辅助量子纠错（EAQEC）码的系列。