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

Abstract

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 $r=tR+1$, $t\ge 1$.