Linear nonbinary covering codes and saturating sets in projective spaces

Let A_{R,q} denote a family of covering codes, in which the covering radius R and the size q of the underlying Galois field are fixed, while the code length tends to infinity. In this paper, infinite sets of families A_{R,q}, where R is fixed but q ranges over an infinite set of prime powers are considered, and the dependence on q of the asymptotic covering densities of A_{R,q} is investigated. It turns out that for the upper limit of the covering density of A_{R,q}, the best possibility is O(q). The main achievement of the present paper is the construction of asymptotic optimal infinite sets of families A_{R,q} for any covering radius R >= 2. We first showed that for a given R, to obtain optimal infinite sets of families it is enough to construct R infinite families A_{R,q}^{(0)},A_{R,q}^{(1)},...,A_{R,q}^{(R-1)} such that, for all u >= u_{0}, the family A_{R,q}^{(v)} contains codes of codimension r_{u}=Ru+v and length f_{q}^{v}(r_{u}) where f_{q}^{v}(r)=O(q^{(r-R)/R) and u_{0} is a constant. Then, we were able to construct the needed families A_{R,q}^{(v)} for any covering radius R >= 2, with q ranging over the (infinite) set of R-th powers. For each of these families A_{R,q}^{(v)}, the lower limit of the covering density is bounded from above by a constant independent of q.