On the duals of quasi-cyclic codes and their application to quantum codes

This paper presents some results on 1-generator quasi-cyclic (QC) codes and their duals over finite fields. We have explicitly obtained a set of generators for the dual of a 1-generator QC code C from the generator of C, for some restricted cases. From the form of the generators of the dual code C⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\perp $$\end{document}, we have derived upper bounds on the minimum distance of C⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\perp $$\end{document}. For their applications in constructing quantum codes, we have presented some criteria for 1-generator QC codes to be self-orthogonal or self-dual. Then using the CSS construction, we have obtained some new and better quantum codes with the help of self-orthogonal 1-generator QC codes.