Linear nonbinary covering codes and saturating sets in projective spaces

Let AR,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. The construction of families with small asymptotic covering densities is a classical problem in the area Covering Codes. In this paper, infinite sets of families AR,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 AR,q is investigated. It turns out that for the upper limit μ∗q(R,AR,q) of the covering density of AR,q , the best possibility is μ ∗ q(R,AR,q) = O(q). (1) The main achievement of the present paper is the construction of optimal infinite sets of families AR,q , that is, sets of families such that (1) holds, 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 (0) R,q ,A (1) R,q , . . . ,A (R−1) R,q such that, for all u ≥ u0, the family A (γ)