Quantum Gabidulin Codes

• Quantum Gabidulin codes are entanglement-assisted quantum error-correcting codes constructed from Gabidulin rank-metric codes using e-Galois inner products and self-dual bases.
• The paper derives a closed-form formula for the e-Galois hull dimension, directly linking it to precise criteria for self-orthogonality and self-duality in the code structure.
• By adjusting the parameter e, the framework enables tunable trade-offs between quantum dimension and entanglement consumption, notably allowing Hermitian self-duality in even characteristic.

Searching arXiv for the cited and related papers to ground the article with current identifiers. Quantum Gabidulin codes are entanglement-assisted quantum error-correcting codes obtained from Gabidulin rank-metric codes over $\mathbb{F}_{q^m}$ by using $e$-Galois inner products, $e$-Galois duality, and the associated hull dimensions. In the formulation developed for Gabidulin codes generated from a self-dual $\mathbb{F}_q$-basis, the $e$-Galois hull can be computed in closed form, which in turn yields exact criteria for linear complementary duality, self-orthogonality, and self-duality, together with explicit EAQECC parameter families (Islam et al., 2022). A notable consequence is that when $q$ and $m$ are even, Gabidulin codes can be self-dual at the Hermitian point $e=m/2$, despite the nonexistence of Euclidean self-dual MRD codes in even characteristic (Islam et al., 2022).

1. Rank-metric and Galois-theoretic framework

Let $q$ be a prime power and let $\mathbb{F}_{q^m}$ be the degree-$e$0 extension of $e$1. For a vector $e$2, the rank weight is the $e$3-dimension of the span of its coordinates, equivalently the rank over $e$4 of an $e$5 matrix representation with respect to a fixed $e$6-basis. The rank distance between $e$7 and $e$8 is the rank weight of $e$9 (Islam et al., 2022).

A Gabidulin code is a linear code $e$0 with parameters $e$1 that is maximum rank distance. For length $e$2 and dimension $e$3, it attains the rank Singleton bound with minimum rank distance

$e$4

With respect to the Hamming metric over $e$5, the same code is also MDS with minimum Hamming distance

$e$6

A standard realization uses linearized polynomials. For an $e$7-basis $e$8 of $e$9,

$\mathbb{F}_q$0

Equivalently, $\mathbb{F}_q$1 is generated by the Moore matrix built from the successive Frobenius powers of the basis elements.

The Galois-theoretic structure enters through the $\mathbb{F}_q$2-Galois inner product, defined for $\mathbb{F}_q$3 by

$\mathbb{F}_q$4

The case $\mathbb{F}_q$5 is the Euclidean inner product, while for even $\mathbb{F}_q$6 the case $\mathbb{F}_q$7 is equivalent to the Hermitian inner product. The corresponding dual code is

$\mathbb{F}_q$8

and the $\mathbb{F}_q$9-Galois hull is

$e$0

Its dimension controls whether the code is $e$1-Galois LCD, self-orthogonal, or self-dual.

2. Self-dual bases and the dual of a Gabidulin code

The explicit theory depends on the choice of basis. If $e$2 is dual to $e$3 under the trace pairing, then

$e$4

When $e$5, the basis is self-dual. Such bases exist if and only if $e$6 is even or both $e$7 and $e$8 are odd, a criterion attributed in the paper to Wan (Islam et al., 2022).

This basis choice is structurally decisive. Self-dual bases diagonalize the trace pairing and align the Moore-matrix description of the code with the $e$9-Galois inner product, so that duals and hulls can be obtained by rank computations on shifted Moore matrices. If $q$0 is dual to $q$1, then the parity-check matrix of $q$2 is built from the first $q$3 Frobenius shifts of $q$4, and a generator matrix for the $q$5-Galois dual is obtained by applying the coordinate-wise Frobenius $q$6-power to that parity-check matrix (Islam et al., 2022).

This mechanism is the algebraic bridge from classical Gabidulin codes to quantum constructions. Once the dual is explicit, the hull becomes computable, and once the hull is computable, entanglement consumption in the associated EAQECC can be read off directly.

3. Closed-form hull dimensions and exact structural criteria

For Gabidulin codes generated from a self-dual basis, Islam and Horlemann obtain an explicit piecewise formula for the $q$7-Galois hull dimension. If $q$8 is a self-dual $q$9-basis and $m$0 is the Gabidulin code of length $m$1 and dimension $m$2, then (Islam et al., 2022)

$m$3

Several structural consequences follow immediately.

First, $m$4 is $m$5-Galois LCD if and only if $m$6. Thus, in this setting, Euclidean duality is the unique Galois specialization yielding trivial hull for all nontrivial $m$7.

Second, assuming $m$8, the code is $m$9-Galois self-orthogonal if and only if either $e=m/2$0 or $e=m/2$1. In other words, self-orthogonality appears precisely when the hull reaches the full code dimension.

Third, if $e=m/2$2 is even and $e=m/2$3, then $e=m/2$4 is $e=m/2$5-Galois self-dual if and only if $e=m/2$6. This identifies the Hermitian point as the unique self-dual point in the self-dual-basis regime.

The resulting contrast with Euclidean self-duality is central. The paper recalls the known fact, due to Nebe and Willems, that no Euclidean self-dual MRD code exists when $e=m/2$7 is even. By contrast, if $e=m/2$8 and $e=m/2$9 are even, a self-dual basis exists, and the formula above shows that for $q$0 and $q$1 one has

$q$2

so Hermitian self-dual Gabidulin codes do exist in even characteristic (Islam et al., 2022).

4. Conversion to entanglement-assisted quantum codes

The quantum construction used here is a hull-to-ebit conversion for classical MDS codes. If $q$3 is a classical MDS code over $q$4 with parameters $q$5, then there exists an EAQECC over $q$6 with parameters

$q$7

so the number of ebits is

$q$8

and the quantum dimension is

$q$9

The minimum distance is inherited from the classical MDS code and satisfies the EAQECC Singleton bound

$\mathbb{F}_{q^m}$0

(Islam et al., 2022).

Applying this lemma to $\mathbb{F}_{q^m}$1 with $\mathbb{F}_{q^m}$2 and $\mathbb{F}_{q^m}$3 gives two explicit EAQECC families.

If $\mathbb{F}_{q^m}$4 and

$\mathbb{F}_{q^m}$5

then there exists an EAQECC with parameters

$\mathbb{F}_{q^m}$6

If $\mathbb{F}_{q^m}$7 and

$\mathbb{F}_{q^m}$8

then there exists an EAQECC with the same parameter form

$\mathbb{F}_{q^m}$9

The significance of this construction lies in parameter flexibility. Earlier Galois-hull-based EAQECC constructions cited in the paper, especially for GRS codes, impose upper bounds on the classical dimension $e$00 depending on $e$01, $e$02, and the extension structure. In the Gabidulin setting treated here, $e$03 is unconstrained in the range $e$04 once a self-dual basis exists, and varying $e$05 adjusts the tradeoff between quantum dimension and entanglement consumption while preserving the classical minimum distance (Islam et al., 2022).

5. Representative parameter regimes

The paper gives explicit examples illustrating how the $e$06-parameter controls the hull and hence the quantum code parameters (Islam et al., 2022).

For $e$07 and $e$08, a self-dual basis of $e$09 over $e$10 exists. Taking $e$11, one has $e$12.

When $e$13, the hull dimension is $e$14, so the EAQECC parameters are

$e$15

This is the Euclidean LCD case.

When $e$16, the hull dimension is

$e$17

so the EAQECC parameters become

$e$18

At the Hermitian point $e$19, the hull dimension is $e$20, giving

$e$21

Here the underlying Gabidulin code is Hermitian self-dual, and the entanglement requirement vanishes.

When $e$22, the second branch applies and

$e$23

again producing

$e$24

The example displays Euclidean LCD behavior at $e$25, reduced entanglement at intermediate $e$26, and symmetry around the Hermitian point.

A second example uses $e$27, $e$28, $e$29, and $e$30. Because $e$31 and $e$32 are both odd, a self-dual basis exists. The Gabidulin code has distance

$e$33

Since $e$34,

$e$35

and the EAQECC parameters are

$e$36

Changing only $e$37 modifies the hull and therefore the quantum dimension and ebit count while keeping $e$38 fixed: for $e$39, one gets $e$40 and

$e$41

whereas for $e$42 one gets $e$43 and

$e$44

The paper also records a large-parameter instance with $e$45, $e$46, $e$47, and $e$48, yielding

$e$49

6. Significance, limitations, and research directions

The main conceptual contribution of this theory is the identification of a new self-dual phenomenon at the Hermitian point. For even $e$50, Euclidean self-dual Gabidulin or MRD codes do not exist, but the $e$51-Galois framework shows that Hermitian self-dual Gabidulin codes do exist whenever $e$52 and $e$53 are even. In quantum terms, this permits zero-ebit constructions at $e$54 that are inaccessible under the Euclidean inner product (Islam et al., 2022).

A second contribution is the exact tunability provided by the hull formula. Since

$e$55

and

$e$56

the piecewise expression for $e$57 gives direct control over entanglement consumption and quantum dimension, while the minimum distance remains $e$58. This separation between distance preservation and hull-driven tunability is the operative mechanism behind the two EAQECC families.

The framework is nevertheless conditional on the existence of a self-dual basis. By Wan’s criterion, this holds exactly when $e$59 is even or both $e$60 and $e$61 are odd. When no self-dual basis exists, the hull dimension depends on the particular basis chosen, and the paper identifies this as an open direction. It also points to further topics such as MacWilliams identities under $e$62-Galois duality and extending explicit hull computations to other MRD code families (Islam et al., 2022).

Within the rank-metric literature, Quantum Gabidulin codes in this sense are therefore best understood as a Galois-hull-based quantum extension of Gabidulin algebra: the rank-metric MRD/MDS structure supplies the distance, the self-dual-basis formalism supplies explicit hull control, and the hull-to-ebit conversion supplies the quantum parameters.