Smallest cyclically covering subspaces of F_qⁿ, and lower bounds in Isbell's conjecture

For a prime power q and a positive integer n, we say a subspace U of F_qⁿ is cyclically covering if the union of the cyclic shifts of U is equal to F_qⁿ. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of F_qⁿ. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed q, we answer the question completely for infinitely many values of n (which take the form of certain geometric series). Our results imply lower bounds for a well-known conjecture of Isbell, and a generalisation theoreof, supplementing lower bounds due to Spiga. We also consider the analogous problem for general representations of groups. We use arguments from combinatorics, representation theory and finite field theory.