Abstract
The Critical Problem in matroid theory is the problem for finding the maximum dimension of a subspace that contains no element of a fixed subset S of Fqk. This problem was posed by H. Crapo and G.-C. Rota and has been one of the significant problems in matroid theory. It can be interpreted in terms of a linear code over a finite field as to find the critical exponent of a linear code. This paper introduces the critical exponents of linear codes over finite chain rings, and extends Kung's upper bound on the critical exponent of a representable matroid over Fq to a linear code over a finite chain ring. Consequently, we present some codes whose critical exponents attain the bound.
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