Lower Bound for the Parity Learning Problem
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).