Clifton–Huang conjecture on hyperplane covers of the hypercube

Let $f(n,k)$ be the minimum size of an almost $k$-cover of $Q^n={0,1}^n$, meaning a set of hyperplanes that avoids the origin and covers every other point at least $k$ times.

Clifton–Huang conjecture. For $k \ge 2$ and $n$ sufficiently large,

$$f(n,k) = n + \binom{k}{2}.$$

The conjecture concerns the large-dimension, fixed-multiplicity regime. The paper notes that Sauermann and Wigderson's algebraic lower bounds are smaller than this conjectured value, so the claim remains open.