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:

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 \(r=tR+1\), \(t\ge 1\).

长度函数\(\ell _q(r,R)\)是具有余维（冗余）r和覆盖半径R的q元线性\([n,nr]_qR\)码的最小可能长度n。令\(s_q(N,\rho )\)为射影空间\(\textrm{PG}(N,q)\)中\(\rho \)饱和集的最小大小。\([n,nr]_qR\)码与\ (\textrm{PG}(r-1,q)中的\((R-1)\)饱和n集之间存在一一对应关系)\)意味着\(\ell _q(r,R)=s_q(r-1,R-1)\)。在这项工作中，对于\(R\ge 3\) ， \(\ell _q(tR+1,R)\)上的新渐近上限通过以下形式获得：

新的界限本质上比已知的界限更好。对于\(t=1\) ，射影空间\(\textrm{PG}(R,q)\)中的\((R-1)\)饱和集的新构造，提供小尺寸的集，被提议。构造得到的\([n,n-(R+1)]_qR\)码具有最小距离\(R+1\)，即几乎是MDS(AMDS)码。这些代码被视为提升结构（所谓的“ \(q^m\) -连接结构”）中的起始代码，用于覆盖代码以获得具有不断增长的余维数\(r=tR+1\)的无限代码族。) , \(t\ge 1\)。