Characterizing a Generator Polynomial Matrix for the Dual of a Multi-Twisted Code

The class of multi-twisted (MT) codes generalizes the classes of cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. We establish the correspondence between MT codes over \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} of index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_q[x]$$\end{document}-submodules of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(\mathbb{F}_q[x]\right)<^>\ell$$\end{document}. Thus, a basis of an MT code exists and is used to build a generator polynomial matrix (GPM). We prove some GPM properties, for example, relationship to code dimension, the identical equation, Hermite normal form. Hence, we prove a GPM formula for the dual code of an MT code. Finally, we obtain the necessary and sufficient conditions for the self-orthogonality and self-duality of MT codes.