  We study the problem of determining the minimum number $f(n,k,d)$ of affine
subspaces of codimension $d$ that are required to cover all points of
$\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the
origin at most $k-1$ times. The case $k=1$ is a classic result of Jamison,
which was independently obtained by Brouwer and Schrijver for $d = 1$. The
value of $f(n,1,1)$ also follows from a well-known theorem of Alon and F\"uredi
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 $k \ge 2^{n-d-1}$ we have
$f(n,k,d)=2^d k - \left \lfloor \frac{k}{2^{n-d}} \right \rfloor$, while for $n
> 2^{2^d k-k-d+1}$ we have $f(n,k,d)= n + 2^dk-d-2$, and also study the
transition 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.
