论文标题
用双含式准循环代码构建量子代码
On construction of quantum codes with dual-containing quasi-cyclic codes
论文作者
论文摘要
量子误差理论的主要目标之一是构建具有最佳参数和属性的量子代码。在本文中，我们提出了一类2代准循环代码，并研究了它们在量子代码在小领域的构建中的应用。首先，确定了这些2转的准循环代码的一些足够条件，以确定有关Hermitian内部产品的双重含量。然后，我们利用这些含有双重的准循环代码通过著名的Hermitian结构来生产量子代码。此外，我们在这些准循环代码的最小距离上提出了一个下限，这有助于构建具有较大长度和尺寸的量子代码。作为计算结果，许多超过量子Gilbert-Varshamov绑定的新量子代码构建在$ f_q $上，其中$ Q $为$ 2,3,4,5 $。特别是，16个二进制量子代码在格拉斯的表\ cite {grassl：codetables}的最小距离上增加了下限。在非二进制情况下，许多量子代码是新的或具有比文献中的参数更好的参数。
One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the construction of quantum codes over small fields. Firstly, some sufficient conditions for these 2-generator quasi-cyclic codes to be dual-containing concerning Hermitian inner product are determined. Then, we utilize these Hermitian dual-containing quasi-cyclic codes to produce quantum codes via the famous Hermitian construction. Moreover, we present a lower bound on the minimum distance of these quasi-cyclic codes, which is helpful to construct quantum codes with larger lengths and dimensions. As the computational results, many new quantum codes that exceed the quantum Gilbert-Varshamov bound are constructed over $F_q$, where $q$ is $2,3,4,5$. In particular, 16 binary quantum codes raise the lower bound on the minimum distance in Grassl's table \cite{Grassl:codetables}. In nonbinary cases, many quantum codes are new or have better parameters than those in the literature.
