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

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