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

The length function \(\ell _q(r,R)\) is the smallest possible length n of a q-ary linear \([n,n-r]_qR\) 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 \(\textrm{PG}(N,q)\). There is a one-to-one correspondence between \([n,n-r]_qR\) codes and \((R-1)\)-saturating n-sets in \(\textrm{PG}(r-1,q)\) that implies \(\ell _q(r,R)=s_q(r-1,R-1)\). In this work, for \(R\ge 3\), new asymptotic upper bounds on \(\ell _q(tR+1,R)\) are obtained in the following form:

$$\begin{aligned}&\bullet ~\ell _q(tR+1,R) =s_q(tR,R-1)\\&\hspace{0.4cm} \le \root R \of {\frac{R!}{R^{R-2}}}\cdot q^{(r-R)/R}\cdot \root R \of {\ln q}+o(q^{(r-R)/R}), \hspace{0.3cm} r=tR+1,~t\ge 1,\\&\hspace{0.4cm}~ q\text { is an arbitrary prime power},~q\text { is large enough};\\&\bullet ~\text { if additionally }R\text { is large enough, then }\root R \of {\frac{R!}{R^{R-2}}}\thicksim \frac{1}{e}\thickapprox 0.3679. \end{aligned}$$

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 \(\textrm{PG}(R,q)\), providing sets of small sizes, is proposed. The \([n,n-(R+1)]_qR\) 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 \(r=tR+1\), \(t\ge 1\).