Finding a matrix of rank n to the o(1/log log n) in a subspace of n by n matrices over F_{2^r} promised to contain a rank-1 matrix is hard, assuming NP has no subexponential algorithms.
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Strong Inapproximability for a Promise Rank Problem
Finding a matrix of rank n to the o(1/log log n) in a subspace of n by n matrices over F_{2^r} promised to contain a rank-1 matrix is hard, assuming NP has no subexponential algorithms.