According to the definition, an $\varepsilon$-universal linear hash function family, given a field $\mathbb{F}$, is a set of linear transformations $\mathcal{H} \subseteq \mathbb{F}^{m,n}$ such that for any $\mathbf{v} \in \mathbb{F}^n \setminus \{0 \}$ if $H \sim U(\mathcal{H})$ $$ \Pr[ H \mathbf{v} = 0 ] \leq \varepsilon. $$ My question is whether it is possible to have an $\varepsilon$-universal linear hash function family with low Hamming weight (over small order field, say $\mathbb{F}_2$), that is, calling $w( \cdot )$ the hamming weight of a vector over $\mathbb{F}^n$, we want to have a constant $M$ such that for "almost any" $H \in \mathcal{H}$ $$ w(H \mathbf{v}) \leq M \cdot w(\mathbf{v}) $$
Now, I've ruled out the possibility to have $M = 1$ as this implies that the column of $M$ need to be vectors of the kind $\alpha \cdot e_i = (0, \ldots, \alpha, \ldots, 0)$ in the $i$-th position and one of the vectors $e_1$, $e_2$ or $e_1 - e_2$ is going to be mapped to zero with non negligible probability. Can one do better? For instance can $M = \Theta(\log(n))$ or $M = \Theta(\sqrt{n})$ be achieveable? Is there any result in the positive direction?
