On the hull of linearized polynomial codes

Abstract

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 $\mathcal{C}(\boldsymbol{\alpha})=\operatorname{im}\Phi_{\boldsymbol{\alpha}}$, with $\Phi_{\boldsymbol{\alpha}}=\sum_i\alpha_iF_i$, we prove the master hull--rank formula $\dim\operatorname{Hull}(\mathcal{C}(\boldsymbol{\alpha}))=\operatorname{rank}(\Phi_{\boldsymbol{\alpha}})-\operatorname{rank}(G(\boldsymbol{\alpha}))$, where $G(\boldsymbol{\alpha})$ is the associated Gram matrix over $\mathbb{F}_q$. Specializing to $C_{\lambda,\mu}=\operatorname{im}(\lambda x+\mu L(x))$, we obtain a quadratic Gram pencil $\lambda^2G_0+\lambda\mu G_1+\mu^2G_2$ 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 $\delta=\dim_{\mathbb{F}_q}(\operatorname{im}\phi_{\lambda,\mu}\cap\ker\phi_{\lambda,\mu}^{\dagger})$, and the extremal condition $\delta=d$ 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.