A new construction of quantum codes from quasi-cyclic codes over finite fields

In this paper we present a construction of quantum codes from 1-generator quasi-cyclic (QC) codes of index 2 over a finite field Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document}. We have studied QC codes of index 2 as a special case of Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document}-double cyclic codes. We have determined the structure of the duals of such QC codes and presented a necessary and sufficient condition for them to be self-orthogonal. A construction of 1-generator QC codes with good minimum distance is also presented. To obtain quantum codes from QC codes, we use the Calderbank-Shor-Steane (CSS) construction. Few examples have been given to demonstrate this construction. Also, we present two tables of quantum codes with good parameters obtained from QC codes over Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document}.