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© 2016, Springer International Publishing. For a subgroup L of the symmetric group (Formula presented.) , we determine the minimal base size of (Formula presented.) acting on (Formula presented.) as an imprimitive linear group. This is achieved by computing the number of orbits of GLd(q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of Vm(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of (Formula presented.) satisfy a conjecture of Pyber concerning bases.