  In Exact Quantum Query model, almost all of the Boolean functions for which
non-trivial query algorithms exist are symmetric in nature. The most well known
techniques in this domain exploit parity decision trees, in which the parity of
two bits can be obtained by a single query. Thus, exact quantum query
algorithms outperforming parity decision trees are rare. In this paper we first
obtain optimal exact quantum query algorithms ($Q_{algo}(f)$) for a direct sum
based class of $\Omega \left( 2^{\frac{\sqrt{n}}{2}} \right)$ non-symmetric
functions. We construct these algorithms by analyzing the algebraic normal form
together with a novel untangling strategy. Next we obtain the generalized
parity decision tree complexity ($D_{\oplus}(f)$) analysing the Walsh Spectrum.
Finally, we show that query complexity of $Q_{algo}$ is $\lceil \frac{3n}{4}
\rceil$ whereas $D_{\oplus}(f)$ varies between $n-1$ and $\lceil \frac{3n}{4}
\rceil+1$ for different classes, underlining linear separation between the two
measures in many cases. To the best of our knowledge, this is the first family
of algorithms beyond generalized parity (and thus parity) for a large class of
non-symmetric functions. We also implement these techniques for a larger
(doubly exponential in $\frac{n}{4}$) class of Maiorana-McFarland type
functions, but could only obtain partial results using similar algorithmic
techniques.
