Unweighted One-Sided Code Sparsifiers and Thin Subgraphs

For a linear code $\mathcal{C} \subseteq \mathbb{F}_2^n$ and $\alpha \in [0,1]$, call a set $S \subseteq [n]$ an (unweighted) one-sided $\alpha$-sparsifier of $\mathcal{C}$ if for all $c \in \mathcal{C}$, $\mathrm{wt}(c_S)\geq \alpha \cdot \mathrm{wt}(c)$, where $c_S$ is the projection of $c$ onto the coordinates in $S$ and $\mathrm{wt}(c)$ is the Hamming weight of $c$. \\ We show that every $k$-dimensional linear code $\mathcal{C}\subseteq \mathbb{F}_2^n$ has at least $2^{n - k}$ many unweighted one-sided $1/2$-sparsifiers and hence one of size at most $n/2 + O(\sqrt{n k})$. As an application, letting $\mathcal{C} \subseteq \mathbb{F}_2^E$ denote the cut-space of a graph $G=(V, E)$, we show a lower bound of $2^{\lvert E \rvert- (\lvert V \rvert - 1)}$ on the number of $1/2$-thin subgraphs of $G$ and the existence of a $1/2$-thin subgraph with at least $\lvert E \rvert /2-O(\sqrt{\lvert E \rvert \cdot \lvert V \rvert})$ edges. In contrast to previous results on thin subgraphs, our proofs are purely "combinatorial".