On the hull of linearized polynomial codes

2026-04-24

AI Summary

This work investigates the hull dimension of linear codes defined by q-polynomial operators to determine the number of pre-shared entangled pairs required for entanglement-assisted quantum error-correcting codes. By employing a unified Gram matrix approach, the authors establish an exact relationship between hull dimension and the rank of the Gram matrix, derive a central hull–rank formula, and characterize LCD (linear complementary dual) trajectories via quadratic Gram pencils. In the case of cyclic Gram matrices, they provide closed-form criteria and a complete classification, incorporating trace isotropy conditions to describe extremal hull dimensions. Leveraging Delsarte duality, circulant matrix theory, and finite-field projective geometry, they precisely count LCD and non-LCD codes, proving that the asymptotic density of LCD codes tends to one as \( q \to \infty \), with explicit examples over \( \mathbb{F}_{64} \) and supporting SageMath code.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

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 $\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα}$, 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.

Problem

• hull dimension
• linearized polynomial codes
• entanglement-assisted quantum error-correcting codes
• LCD codes
• Gram matrix

Innovation

• Gram matrix
• hull dimension
• linearized polynomial codes
• LCD codes
• rank-distance codes