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Aug 5, 2015 at 13:29 comment added Will Sawin @GerryMyerson Let me say instead that Weil does not immediately give you a lower bound. I't easy to see that you can get it down to $q^{n-dm}$ using different norm polynomials in different subsets of the variables. So there is still some room between this lower bound and the explicit examples.
Aug 4, 2015 at 23:28 comment added Gerry Myerson The question asks for $q$ much larger than $d$, but much smaller than $n$, so it seems $d=n$ does not apply.
Aug 4, 2015 at 20:37 history answered Will Sawin CC BY-SA 3.0