Learning k-parities with Membership Queries and Persistent Noise Random independent misclassification error is an inappropriate noise model for a membership query (MQ) oracle because for any noise rate $\eta&lt;1/2$ one can eliminate noise to an arbitrary extent by simply querying the oracle $p(1/(1-2\eta))$ times with the same input for some polynomial $p(\cdot)$ and taking the majority label returned.  

The random persistent misclassification noise model of Goldman, Kearns, and Schapire, plugs this "loophole" by stipulating that an oracle that assigns a label $l$ to some sample $x$, always assigns label $l$ to sample $x$.

My question is: What are the known best upper bounds on the time and queries required to solve the learning parities problem with membership queries and random persistent misclassification noise for some constant value of $k$ (the number of relevant attributes)? Is the problem known to be solvable in $O(n \log (n/\delta))$ time and $O(\log (n/\delta))$ queries w.r.t. $n$, the number of attributes and $\delta$, the probability of error?
If I understand correctly, this is exactly equivalent to local decoding of the Hadamard code.

We can think of the oracle (with the noise) as being chosen beforehand, and after that we can view the oracle as a corrupted codeword of the Hadamard code. Learning the parity in this case is exactly the task of local decoding the original message from the corrupted codeword.

This is well-known to be possible using $O(n \log n/\delta)$ queries, using the Goldreich-Levin theorem, e.g.:
https://lucatrevisan.wordpress.com/2009/03/09/cs276-lecture-12-goldreich-levin/

On the other hand, it clearly can not be done using $o(n)$ queries, since the algorithm has learn $n$ bits about the parity - i.e., its attributes.