We study the approximability of predicates on k variables from a
domain [q], and give a new sufficient condition for such predicates
to be approximation resistant under the Unique Games Conjecture.
Specifically, we show that a predicate P is approximation resistant
if there exists a balanced pairwise independent distribution over
[q]^k whose support is contained in the set of satisfying assignments
to P.

Using constructions of pairwise indepenent distributions this result
implies that:

For general k \ge 3 and q \ge 2, the Max k-CSP_q problem is
UG-hard to approximate within q^{\lceil \log_2 k +1 \rceil - k} + \epsilon.

For k \geq 3 and q prime power, the hardness ratio is improved to
kq(q-1)/q^k + \epsilon.

For the special case of q = 2, i.e., boolean variables, we can
sharpen this bound to (k + O(k^{0.525}))/2^k + \epsilon, improving
upon the best previous bound of 2k/{2^k} + \epsilon (Samorodnitsky
and Trevisan, STOC'06) by essentially a factor 2.

Finally, for q=2, assuming that the famous Hadamard Conjecture is
true, this can be improved even further, and the O(k^{0.525}) term
can be replaced by the constant 4.