On the Covering Dimension of a Linear Code

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 and, thus, generalizes invariants that lie at the heart of several fundamental problems in a coding theory. An upper bound on the covering dimension is proved, improving Kung's classical bound for the critical exponent. In addition, a construction is given for linear codes that attain equality in this covering dimension bound.