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 ${\mathbb{F}}_q^k$. 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 ${\mathbb{F}}_q$ to a linear code over a finite chain ring. Consequently, we present some codes whose critical exponents attain the bound.

拟阵理论中的关键问题是寻找不包含固定子集S的元素的子空间的最大维数的问题${\mathbb{F}}_q^{克}$. 这个问题是由 H. Crapo 和 G.-C. 提出的。Rota 和 一直是拟阵理论中的重大问题之一。可以将其解释为有限域上的线性代码，以找到线性代码的临界指数。本文介绍了有限链环上线性码的临界指数，并在可表示拟阵的临界指数上扩展了 Kung 上界${\mathbb{F}}_q$到有限链环上的线性代码。因此，我们提出了一些关键指数达到界限的代码。