论文标题
关于线性代码和新的纠缠量子误差校正代码的GALOIS船体
On Galois hulls of linear codes and new entanglement-assisted quantum error-correcting codes
论文作者
论文摘要
线性代码的Galois船体是自身及其Galois双重代码的交集，这在这些年来引起了研究人员的兴趣。在本文中，我们研究线性代码的Galois船体。首先，发现线性代码的Galois船体尺寸的对称性。线性代码的一些新的必要条件是Galois自动执行代码，GALOIS自动偶数代码和Galois线性互补双码代码。然后，我们提出了明确的方法来构建从给定的Galois自动实行代码的较大长度的GALOIS自动执行代码。作为一个应用，进一步得出了具有任意维度的Galois船体的较大长度的线性代码。还构建了两种新类别的Hermitian自动距离最大距离（MDS）代码的新类别。最后，将所有结果应用于纠缠辅助量子错误校正校正代码（EAQECC），许多新的$ Q $ - ary或$ \ sqrt {q} $ - Ary EaqeCCS和MDS EAQECCS和MDS EAQECC的速率大于或等于或等于$ \ frac {1} $ frac {1} $ and-agerate $ and oftains $ and oftauty。此外，许多$ \ sqrt {q} $ - 长度$ n> \ sqrt {q}+1 $的最小距离大于或等于$ \ lceil \ frac {\ frac {\ sqrt {q}}} {2} {2} \ rceil $。
The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the dimensions of Galois hulls of linear codes is found. Some new necessary and sufficient conditions for linear codes being Galois self-orthogonal codes, Galois self-dual codes, and Galois linear complementary dual codes are characterized. Then, we propose explicit methods to construct Galois self-orthogonal codes of larger length from given Galois self-orthogonal codes. As an application, linear codes of larger length with Galois hulls of arbitrary dimensions are further derived. Focusing on the Hermitian inner product, two new classes of Hermitian self-orthogonal maximum distance separable (MDS) codes are also constructed. Finally, applying all the results to the construction of entanglement-assisted quantum error-correcting codes (EAQECCs), many new $q$-ary or $\sqrt{q}$-ary EAQECCs and MDS EAQECCs with rates greater than or equal to $\frac{1}{2}$ and positive net rates can be obtained. Moreover, the minimum distance of many $\sqrt{q}$-ary MDS EAQECCs of length $n>\sqrt{q}+1$ is greater than or equal to $\lceil \frac{\sqrt{q}}{2} \rceil$.
