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

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}&(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\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}&(b)\; \ell_q(r,R)< 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\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} .