The critical exponent of a matroid is one of the important parameters in
matroid theory and is related to the Rota and Crapo's Critical Problem. This
paper introduces the covering dimension of a linear code over a finite field,
which is analogous to the critical exponent of a representable matroid. An
upper bound on the covering dimension is conjectured and nearly proven,
improving a classical bound for the critical exponent. Finally, a construction
is given of linear codes that attain equality in the covering dimension bound