An upper bound for the complexity of linearized coverings in a finite field

The minimal number of systems of linear equations with $n$ unknowns over a finite field $F_q$, such that the union of all solutions of the systems forms an exact cover for a given subset in $F_q^n$, is the complexity of a linearized covering. An upper bound for the complexity for “almost all” subsets in $F_q^n$ is presented.