Construction of self-dual MDR cyclic codes over finite chain rings

Maximum distance with respect to rank codes, or MDR codes, are a family of optimal linear codes that meet a Singleton-like bound in terms of the length and rank of the codes. In this paper, we study the construction of self-dual MDR cyclic codes over a finite chain ring R. We present a new form for the generator polynomials of cyclic codes over R of length n with the condition that the length n and the characteristic of R are relatively prime. Consequently, sufficient and necessary conditions for cyclic codes over R to be self-dual and self-orthogonal are obtained. As a result, self-dual MDR cyclic codes over the Galois ring GR(pt,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {GR}(p^t,m)$$\end{document} with length n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} dividing pm-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p^m}-1$$\end{document} are constructed by using torsion codes.