Lower Bound for the Parity Learning Problem What are known lower bounds for the time and query complexity of the problem of learning parities with an adaptive membership query oracle? To be clear the concept space $C$ is $\{x\in \{0,1\}^n \, \, | \, \,\sum_{i=1}^n x_i = k \}$, and the hypothesis space $\mathcal H$ is $\{0,1\}^n$. Given a bitstring of length $n$ as input, the adaptive membership query oracle returns 1 if the parity of the $k$ effective bits is odd and 0 otherwise.
The question is about recovering with membership queries for which it is well known that O( k log n) queries suffice (via BCH codes see Hoffmeister's paper or my COLT 2005 paper). In terms of lower bounds \Omega(k log n) trivially follows from counting the number of parities and using the fact that they are all completely uncorrelated (basic packing-based lower bound).