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 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 $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$, while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$, 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.

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$, and let $\phi \,:\, \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $\phi$ as

\begin{equation*} {\sf avgStretch}(\phi ) = {\mathbb E}[{{\sf dist}}(\phi (x),\phi (x'))], \end{equation*}

$x,x' \in \{0,1\}^{n-1}$

In this paper, we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results.

• For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $\phi _A \,:\, \{0,1\}^{n-1} \to A$ such that ${\sf avgStretch}(\phi _A) = O\left(\sqrt{n}\right)$.

• For $n = 3^k$ let ${A_{\textsf{rec-maj}}} = \{x \in \{0,1\}^n \,:\,{\textsf{rec-maj}}(x) = 1\}$, where ${\textsf{rec-maj}} \,:\, \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3’s. There exists a bijection $\phi _{{\textsf{rec-maj}}} \,:\, \{0,1\}^{n-1} \to{A_{\textsf{rec-maj}}}$ such that ${\sf avgStretch}(\phi _{{\textsf{rec-maj}}}) = O(1)$.

• Let ${A_{{\sf tribes}}} = \{x \in \{0,1\}^n \,:\,{\sf tribes}(x) = 1\}$. There exists a bijection $\phi _{{\sf tribes}} \,:\, \{0,1\}^{n-1} \to{A_{{\sf tribes}}}$ such that ${\sf avgStretch}(\phi _{{\sf tribes}}) = O(\!\log (n))$.

These results answer the questions raised by Benjamini, Cohen, and Shinkar (Isr. J. Math 2016).