<p>An (<i>n</i>,&#xa0;<i>R</i>)-covering sequence is a cyclic sequence whose consecutive <i>n</i>-tuples form a code of length <i>n</i> and covering radius <i>R</i>. Using several construction methods improvements of the upper bounds on the length of such sequences for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n \le 20\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>20</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1 \le R \le 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>R</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, are obtained. The definition is generalized in two directions. An (<i>n</i>,&#xa0;<i>m</i>,&#xa0;<i>R</i>)-covering sequence code is a set of cyclic sequences of length <i>m</i> whose consecutive <i>n</i>-tuples form a code of length&#xa0;<i>n</i> and covering radius <i>R</i>. The definition is also generalized to arrays in which the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(m \times n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> sub-matrices form a covering code with covering radius <i>R</i>. We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.</p>