As part of a research project on derandomization of linear threshold functions I am working on, I am trying to understand the following problem:

Is there a small (polynomial rather than exponential) family of hash functions from $[k]$ to $[n]$, where $k \ll n$, say, $n^{\epsilon}$, which guarantee a "high inner product" with constant probability for every vector in $\mathbb R^n$?

The term "high inner product" should be interpreted as follows: the result of the hash function represents a vector in the binary hypercube, with $1$ in every bucket which a coordinate of $[k]$ is mapped into, and $-1$ elsewhere. Given a vector $v \in \mathbb R^n$, I would like at least a small constant factor of my hash functions to have an inner product which is above the expectation.

Is this possible? Does someone here have an idea for some pointers?