In this paper, we introduce double Quadratic Residue Codes (QRC) of length n=p+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=p+q$$\end{document} for prime numbers p and q in the ambient space F2p×F2q.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {F}}} _{2}^{p}\times {{\mathbb {F}}}_{2}^{q}.$$\end{document} We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when p=q.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=q.$$\end{document} We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting.

中文翻译：

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双二次余数码和自双双循环码

$$\end{document} 我们给出了这个字母表上的可分离和不可分离双二维码的结构，我们证明了这个空间中有趣的双二维码只存在于 p=q.\documentclass[12pt]{minimal } \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document }$$p=q.$$\end{document} 我们给出了这些代码的主要属性，例如它们的幂等生成器和它们的对偶。我们将这些代码与环上的代码联系起来，并展示如何使用它们来构建有趣的格子。作为这些代码的应用，我们提供了自对偶、形式自对偶和最优双 QRC 的示例。我们还提供了在此设置中源自双 QRC 的最著名量子代码的示例。