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The length function   \begin{document}$ \ell_q(r,R) $\end{document}   is the smallest length of a   \begin{document}$  q  $\end{document}  -ary linear code with codimension (redundancy)   \begin{document}$ r $\end{document}   and covering radius   \begin{document}$ R $\end{document}  . In this work, new upper bounds on   \begin{document}$ \ell_q(tR+1,R) $\end{document}   are obtained in the following forms:\begin{document}$ \begin{equation*} \begin{split}&amp;amp;(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\;   R\ge3,\; r = tR+1,\; t\ge1,\\  &amp;amp;\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q.   \end{split} \end{equation*} $\end{document}\begin{document}$ \begin{equation*} \begin{split}&amp;amp;(b)\; \ell_q(r,R)&amp;lt; 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\;   R\ge3,\; r = tR+1,\; t\ge1,\\  &amp;amp;\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}.   \end{split} \end{equation*} $\end{document}In the literature, for   \begin{document}$ q = (q')^R $\end{document}   with   \begin{document}$ q' $\end{document}   a prime power, smaller upper bounds are known; however, when   \begin{document}$ q $\end{document}   is an arbitrary prime power, the bounds of this paper are better than the known ones.  For   \begin{document}$ t = 1 $\end{document}  , we use a one-to-one correspondence between   \begin{document}$ [n,n-(R+1)]_qR $\end{document}   codes and   \begin{document}$ (R-1) $\end{document}  -saturating   \begin{document}$ n $\end{document}  -sets in the projective space   \begin{document}$ \mathrm{PG}(R,q) $\end{document}  . A new construction of such saturating sets providing sets of small size is proposed. Then the   \begin{document}$ [n,n-(R+1)]_qR $\end{document}   codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "  \begin{document}$ q^m $\end{document}  -concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension   \begin{document}$ r = tR+1 $\end{document}  ,   \begin{document}$ t\ge1 $\end{document}  .

Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $