论文标题
Gabidulin代码的GALOIS船体尺寸
Galois Hull Dimensions of Gabidulin Codes
论文作者
论文摘要
对于Prime Power $ Q $，整数$ M $和$ 0 \ leq E \ leq M-1 $，我们研究Gabidulin codes $ g_k（\boldsymbolα）的$ e $ -galois hull hull dimension $ g_k（\boldsymbolα）长度$ m $的$ m $和dimension $ k $ k $ a $ k $ ive $ k $ a $ k $ ive $ \ nathbb {f} $ {f} $}使用$ \ mathbb {f} _ {q^m} $的$ \boldsymbolα$在$ \ mathbb {f} _q $上，我们首先明确计算$ g_k（\boldsymbolα）$的船体维度。然后将提供$ g_k（\boldsymbolα）$的必要条件，即线性互补双（LCD），自动和自dual。我们证明存在$ e $ -galois（其中$ e = \ frac {m} {2} $）什至$ q $的长度$ m $的自dual gabidulin代码，这与已知的事实形成鲜明对比：欧几里得自dual gabidulin codes在$ q $中都不存在。作为一个应用程序，我们构建了两类纠缠的量子误差校正代码（EAQECC），其参数与在这种情况下的已知代码相比具有更大的灵活性。
For a prime power $q$, an integer $m$ and $0\leq e\leq m-1$ we study the $e$-Galois hull dimension of Gabidulin codes $G_k(\boldsymbolα)$ of length $m$ and dimension $k$ over $\mathbb{F}_{q^m}$. Using a self-dual basis $\boldsymbolα$ of $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$, we first explicitly compute the hull dimension of $G_k(\boldsymbolα)$. Then a necessary and sufficient condition of $G_k(\boldsymbolα)$ to be linear complementary dual (LCD), self-orthogonal and self-dual will be provided. We prove the existence of $e$-Galois (where $e=\frac{m}{2}$) self-dual Gabidulin codes of length $m$ for even $q$, which is in contrast to the known fact that Euclidean self-dual Gabidulin codes do not exist for even $q$. As an application, we construct two classes of entangled-assisted quantum error-correcting codes (EAQECCs) whose parameters have more flexibility compared to known codes in this context.
