  We present several families of total boolean functions which have exact
quantum query complexity which is a constant multiple (between 1/2 and 2/3) of
their classical query complexity, and show that optimal quantum algorithms for
these functions cannot be obtained by simply computing parities of pairs of
bits. We also characterise the model of nonadaptive exact quantum query
complexity in terms of coding theory and completely characterise the query
complexity of symmetric boolean functions in this context. These results were
originally inspired by numerically solving the semidefinite programs
characterising quantum query complexity for small problem sizes. We include
numerical results giving the optimal success probabilities achievable by
quantum algorithms computing all boolean functions on up to 4 bits, and all
symmetric boolean functions on up to 6 bits.
