Linear covering codes over nonbinary finite fields

For a prime power q and for integers R, η with R > 0, 0 ≤ η ≤ R − 1, let A R,q = (Cni)i denote an infinite sequence of q-ary linear [ni, ni − ri]qR codes Cni with covering radius R and such that the following two properties hold: (a) the codimension ri = Rti + η, where (ti)i is an increasing sequence of integers; (b) the length ni of Ci coincides with f (η) q (ri), where f (η) q is an increasing function. In this paper, sequences A R,q with asymptotic covering density bounded from above by a constant independent of q are constructed for an arbitrary R, and for each value of η ∈ {0, 1, . . . , R − 1}, under the condition that q = (q′)R. The key tool is the description of new small saturating sets in projective spaces over finite fields, which are the starting point for the q-concatenating constructions of covering codes. A new concept of N -fold strong blocking set is introduced. Several upper bounds on the length function of covering codes and on the smallest sizes of saturating sets are improved.