What are the hardness results known for CSP over $\mathbb{F}_q$?
Here is a summary of what is known about approximability of $k$-CSP over a domain of size $q$:


The best known approximation algorithms for the problem give an $\Omega(q \max(k, \log q)/q^k)$ approximation [MM14 and MNT16].
For $k = \Omega(q)$, there is a matching hardness of $O(kq/q^k)$ by Håstad (UGC-hardness) and Chan (NP-hardness) [Chan13].
For $k$ between $c \log q  /  \log \log q$ and $q$, the best known NP-hardness result is $O(q^2/q^k)$ (follows from [Chan13]).
For $2 &lt; k &lt; c \log q  /  \log \log q$, the best known UGC-hardness is $2^{O(k \log k)} q \, (\log q)^{k/2}/q^k$ by Manurangsi et al. [MNT16].
For $k=2$, the best NP-hardness is $O(\log q/\sqrt{q})$ by Chan [Chan13], the best UGC-hardness is $O(\log q/q^k)$ by Khot et al. [KKMO07]


[Chan13] Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the Symposium on Theory of Computing, pages 447–456, 2013.

[KKMO07] Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproxima bility results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput., 37(1):319–357, 2007.

[MM14] Konstantin Makarychev and Yury Makarychev. Approximation algorithm for non-Boolean Max $k$-CSP. Theory of Computing, 10(13):341–358, 2014.

[MNT16] Pasin Manurangsi, Preetum Nakkiran, and Luca Trevisan. Near-optimal UGC-hardness of approximating Max $k$-CSP$_r$ . In Proceedings of the Workshop on Approximation Algorithms for Combinatorial Optimization Problems (to appear), 2016.