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