Hermitian hull-variation of vector rank-metric codes and self-orthogonal generalized Gabidulin codes

Technical Summary: Hermitian Hull-Variation of Vector Rank-Metric Codes and Self-Orthogonal Generalized Gabidulin Codes

Problem Statement
This paper investigates the "hull-variation problem" for vector rank-metric codes equipped with the Hermitian inner product. The hull of a linear code $C$, defined as $H(C) = C \cap C^\perp$, is a fundamental subcode whose dimension dictates the feasibility of certain constructions, such as entanglement-assisted quantum error-correcting codes (EAQECC). While the hull-variation problem for Euclidean and Hermitian codes in the Hamming metric has been largely resolved (showing that codes can often be transformed into equivalent codes with any prescribed hull dimension up to the original), the analogous problem for vector rank-metric codes remained open. Specifically, the paper addresses whether every vector rank-metric code is equivalent to a Hermitian LCD (Linear Complementary Dual) code (i.e., a code with a trivial hull) and whether Maximum Rank Distance (MRD) codes can be constructed with any prescribed Hermitian hull dimension.

Methodology and Key Contributions

1. Hermitian Hull-Variation Theorem:
   The authors establish that for vector rank-metric codes over $\mathbb{F}_{q^{2m}}$, the Hermitian hull dimension can be reduced by one within the code's equivalence class, provided specific conditions are met.

   • General Case ($n \ge 3$ or $q \ge 4$): The proof utilizes a constructive approach involving generator matrices. By identifying a non-zero vector in the hull and applying a specific equivalence transformation $M \in GL_n(\mathbb{F}_q)$, the authors show that the rank of the matrix $GG^\dagger$ increases by one, thereby reducing the hull dimension by one. This relies on Lemma 3.1, which demonstrates how adding a rank-one term $\lambda aa^\dagger$ to a Hermitian matrix reduces its nullity under specific conditions.
   • Small Parameters ($n=2$): The case $n=2$ is treated separately. For $q \ge 4$, the general construction applies. For $q=3$, a specific argument using quadratic polynomials over $\mathbb{F}_3$ is used to find a transformation that eliminates the hull.
   • Obstruction: The paper identifies a single exceptional case where hull variation is impossible: $(q, n) = (2, 2)$. In this specific instance, the authors provide a counterexample showing that a code with a non-trivial hull cannot be transformed into an equivalent Hermitian LCD code.

2. Scaled Trace-Self-Dual Basis:
   To address the existence of MRD codes with prescribed hull dimensions, the authors introduce the concept of a scaled trace-self-dual basis.

   • Previous work established that ordinary trace-self-dual bases for $\mathbb{F}_{q^{2m}}/\mathbb{F}_q$ exist only when $q$ is even. This left the case of odd $q$ open for constructing Hermitian self-orthogonal Gabidulin codes.
   • The authors define a pair $(\alpha, \lambda)$ where $\alpha$ is a basis and $\lambda \in \mathbb{F}_{q^{2m}}^*$ such that $\text{Tr}(\lambda \alpha_i \alpha_j) = \delta_{ij}$.
   • They prove that such a basis exists for every prime power $q$ and integer $m$. The proof involves selecting $\lambda$ such that its norm equals the inverse of the determinant of the Gram matrix of the trace form, thereby transforming the bilinear form into a standard orthonormal form.

3. Construction of Self-Orthogonal MRD Codes:
   Using the scaled trace-self-dual basis, the authors construct generalized Gabidulin codes that are Hermitian self-orthogonal.

   • They demonstrate that for any $1 \le k \le m$, the generalized Gabidulin code $G_{k,s}(\alpha)$ generated by a scaled trace-self-dual basis is Hermitian self-orthogonal.
   • This construction resolves the open case for odd $q$, allowing for the creation of Hermitian self-dual MRD codes (where $k=m$) for all prime powers.

Main Results

• Hull-Variation Classification: For every pair $(q, n) \neq (2, 2)$, every vector rank-metric code with Hermitian hull dimension $h$ is equivalent to a code with hull dimension $\ell$ for any $0 \le \ell \le h$. Consequently, every such code is equivalent to a Hermitian LCD code.
• Existence of MRD Codes with Prescribed Hull: For every $(q, m) \neq (2, 1)$, every dimension $1 \le k \le m$, and every target hull dimension $0 \le \ell \le k$, there exists an MRD code in $\mathbb{F}_{q^{2m}}^{2m}$ with parameters $[2m, k, 2m-k+1]_{q^{2m}/q}$ and Hermitian hull dimension $\ell$.
• Euclidean Comparison: The paper extends the hull-variation results to the Euclidean inner product for vector rank-metric codes, completing cases left open in previous literature. However, it notes a significant divergence: unlike the Hermitian setting, Euclidean self-orthogonal MRD codes do not exist in even characteristic.

Significance
The paper provides a comprehensive solution to the hull-variation problem for vector rank-metric codes under the Hermitian inner product, identifying the unique obstruction at $(2, 2)$. By introducing the scaled trace-self-dual basis, the authors remove the parity restriction on $q$ that previously hindered the construction of Hermitian self-orthogonal Gabidulin codes. This unifies the theory, ensuring that MRD codes with any admissible Hermitian hull dimension can be constructed for all prime powers. These results are directly motivated by and applicable to the construction of EAQECCs, where the hull dimension determines the number of pre-shared entangled pairs required.