Trace dual of additive cyclic codes over finite fields

In (Shi et al. Finite Fields Appl.80, 102087 2022) studied additive cyclic complementary dual codes with respect to trace Euclidean and trace Hermitian inner products over the finite field F4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_4$$\end{document}. In this article, we extend their results over Fq2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q<^>2},$$\end{document} where q is an odd prime power. We describe the algebraic structure of additive cyclic codes and obtain the dual of a class of these codes with respect to the trace inner products. We also use generating polynomials to construct several examples of additive cyclic codes over F9.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_9.$$\end{document} These codes are better than linear codes of the same length and size. Furthermore, we describe the subfield codes and the trace codes of these codes as linear cyclic 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}.