New bounds for covering codes of radius 3 and codimension 3t+1

The smallest possible length of a $q$-ary linear code of covering radius $R$ and codimension (redundancy) $r$ is called the length function and is denoted by $\ell_q(r,R)$. In this work, for $q$ \emph{an arbitrary prime power}, we obtain the following new constructive upper bounds on $\ell_q(3t+1,3)$: $\ell_q(r,3)\lessapprox \sqrt[3]{k}\cdot q^{(r-3)/3}\cdot\sqrt[3]{\ln q},~r=3t+1, ~t\ge1, ~ q\ge\lceil\mathcal{W}(k)\rceil, 18 <k\le20.339,~\mathcal{W}(k)\text{ is a decreasing function of }k ;$ $\ell_q(r,3)\lessapprox \sqrt[3]{18}\cdot q^{(r-3)/3}\cdot\sqrt[3]{\ln q},~r=3t+1,~t\ge1,~ q\text{ large enough}.$ For $t = 1$, we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space $\mathrm{PG}(3,q)$. A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called ``$q^m$-concatenating constructions'') to obtain infinite families of codes with radius 3 and growing codimension $r = 3t + 1$, $t\ge1$. The new bounds are essentially better than the known ones.