Abstract
An (n, R)-covering sequence is a cyclic sequence whose consecutive n-tuples form a code of length n and covering radius R. Using several construction methods improvements of the upper bounds on the length of such sequences for n <= 20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \le 20$$\end{document} and 1 <= R <= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le R \le 3$$\end{document}, are obtained. The definition is generalized in two directions. An (n, m, R)-covering sequence code is a set of cyclic sequences of length m whose consecutive n-tuples form a code of length n and covering radius R. The definition is also generalized to arrays in which the mxn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \times n$$\end{document} sub-matrices form a covering code with covering radius R. We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.