Subspace coverings with multiplicities

We study the problem of determining the minimum number of affine subspaces of codimension that are required to cover all points of at least times while covering the origin at most times. The case is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for. The value of also follows from a well-known theorem of Alon and FÜredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for we have, while for we have, and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.