Almost covering all the layers of hypercube with multiplicities

Given a hypercube ${\mathcal{Q}}^n:=\{0,1{\}}^n$ in ${\mathbb{R}}^n$ and $k\in \{0,\dots ,n\}$, the $k$-th layer ${\mathcal{Q}}_k^n$ of ${\mathcal{Q}}^n$ denotes the set of all points in ${\mathcal{Q}}^n$ whose coordinates contain exactly $k$ many ones. For a fixed $t\in \mathbb{N}$ and $k\in \{0,\dots ,n\}$, let $P\in \mathbb{R}[x_1,\dots ,x_n]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of ${\mathcal{Q}}^n\setminus {\mathcal{Q}}_k^n$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of ${\mathcal{Q}}_k^n$. In this short note, we show that $$deg(P)\ge \max \{k,n-k\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_1,\dots ,H_m$ in ${\mathbb{R}}^n$, where $m=\max \{k,n-k\}+2t-2$, such that every point of ${\mathcal{Q}}_k^n$ will be covered exactly $t-1$ times, and every other point of ${\mathcal{Q}}^n$ will be covered at least $t$ times. Note that putting $k=0$ and $t=1$, we recover the much celebrated covering result of Alon and Füredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube ${\mathcal{Q}}^n$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.