Motivated by entanglement-assisted quantum error-correcting codes, where the hull dimension determines the number of required pre-shared entangled pairs, we study hulls of two families of $\mathbb{F}_q$-linear codes defined by $q$-polynomial operators over $\mathbb{F}_{q^m}$. Our main tool is a unified Gram-matrix method. For image codes $\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα}$, with $Φ_{\boldsymbolα}=\sum_iα_iF_i$, we prove the master hull--rank formula $\dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ_{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα))$, where $G(\boldsymbolα)$ is the associated Gram matrix over $\mathbb{F}_q$. Specializing to $C_{λ,μ}=\operatorname{im}(λx+μL(x))$, we obtain a quadratic Gram pencil $λ^2G_0+λμG_1+μ^2G_2$ whose determinant describes the LCD locus in $\mathbb{P}^1(\mathbb{F}_q)$. We also treat $\mathbb{F}_{q^m}$-linear rank-distance codes $\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}}$ with the Delsarte inner product, where a $k\times k$ Gram matrix over $\mathbb{F}_{q^m}$ determines the hull dimension. For $L(X)=X^{q^k}$, with $d=\gcd(k,m)$, the resulting circulant Gram matrices yield a closed-form discriminant and a complete classification in three of the four bijectivity configurations over $\mathbb{P}^1(\mathbb{F}_{q^m})$. In the remaining case, the hull dimension equals $δ=\dim_{\mathbb{F}_q}(\operatorname{im}φ_{λ,μ}\cap\kerφ_{λ,μ}^{\dagger})$, and the extremal condition $δ=d$ is characterized by an explicit trace-isotropy criterion. We conclude with an exact count of LCD and non-LCD points, showing that the LCD density tends to $1$ as $q\to\infty$, together with a worked example over $\mathbb{F}_{64}$ and a SageMath verification.

翻译：受纠缠辅助量子纠错码的启发（其中壳维度决定所需的预共享纠缠对数量），我们研究了由$\mathbb{F}_{q^m}$上的$q$-多项式算子定义的两族$\mathbb{F}_q$-线性码的壳。我们的主要工具是统一的格拉姆矩阵方法。对于像码$\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα}$（其中$Φ_{\boldsymbolα}=\sum_iα_iF_i$），我们证明了主壳-秩公式$\dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ_{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα))$，其中$G(\boldsymbolα)$是$\mathbb{F}_q$上的关联格拉姆矩阵。特例化为$C_{λ,μ}=\operatorname{im}(λx+μL(x))$时，我们得到二次格拉姆束$λ^2G_0+λμG_1+μ^2G_2$，其行列式描述了$\mathbb{P}^1(\mathbb{F}_q)$中的LCD轨迹。我们还处理了具有Delsarte内积的$\mathbb{F}_{q^m}$-线性秩距码$\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}}$，其中$\mathbb{F}_{q^m}$上的$k\times k$格拉姆矩阵决定了壳维度。对于$L(X)=X^{q^k}$（其中$d=\gcd(k,m)$），生成的循环格拉姆矩阵可导出闭合形式判别式，并在$\mathbb{P}^1(\mathbb{F}_{q^m})$上的四种双射配置中的三种中实现完全分类。在剩余情况下，壳维度等于$δ=\dim_{\mathbb{F}_q}(\operatorname{im}φ_{λ,μ}\cap\kerφ_{λ,μ}^{\dagger})$，极值条件$δ=d$由显式的迹-各向同性准则刻画。最后我们给出LCD与非LCD点的精确计数，表明当$q\to\infty$时LCD密度趋于$1$，并附带$\mathbb{F}_{64}$上的算例及SageMath验证。