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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2604.23097 |
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| _version_ | 1866913061753847808 |
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| author | Bartoli, Daniele Grimaldi, Giovanni Giuseppe Stănică, Pantelimon |
| author_facet | Bartoli, Daniele Grimaldi, Giovanni Giuseppe Stănică, Pantelimon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23097 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the hull of linearized polynomial codes Bartoli, Daniele Grimaldi, Giovanni Giuseppe Stănică, Pantelimon Information Theory Number Theory 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. |
| title | On the hull of linearized polynomial codes |
| topic | Information Theory Number Theory |
| url | https://arxiv.org/abs/2604.23097 |