Further results on covering codes with radius R and codimension tR + 1

The length function l_q(r,R) is the smallest possible length n of a q-ary linear [n,n-r]_q R code with codimension (redundancy) r and covering radius R. Let s_q(N,\rho) be the smallest size of a \rho-saturating set in the projective space PG(N,q). There is a one-to-one correspondence between [n,n-r]_qR codes and (R-1)-saturating n-sets in PG(r-1,q) that implies l_q(r,R)=s_q(r-1,R-1). In this work, for R\ge3, new asymptotic upper bounds on l_q(tR+1,R) are obtained in the following form: \begin{align*} &\bullet~\ell_q(tR+1,R) =s_q(tR,R-1)\le \sqrt[R]{\frac{R!}{R^{R-2}}}\cdot q^{(r-R)/R}\cdot\sqrt[R]{\ln q}+o(q^{(r-R)/R}),\\ &\hspace{1cm}r=tR+1,~t\ge1,~ q\text{ is an arbitrary prime power},~q\text{ is large enough};\\ &\bullet~\text{ if additionally }R\text{ is large enough, then }\sqrt[R]{\frac{R!}{R^{R-2}}}\thicksim\frac{1}{e}\thickapprox0.3679. \end{align*} The new bounds are essentially better than the known ones. For t=1, a new construction of (R-1)-saturating sets in the projective space PG(R,q), providing sets of small sizes, is proposed. The [n,n-(R+1)]_q R codes, obtained by the construction, have minimum distance R + 1, i.e. they are almost MDS (AMDS) codes. These codes 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.