An important question in the construction of orthogonal arrays is what the minimal size of an array is when all other parameters are fixed. We will introduce a generalization of an inequality for symmetric orthogonal arrays developed by Bierbrauer to the asymmetric case. We will outline a proof that utilizes his algebraic approach and provide several examples for parameter combinations for which this new bound is better than the classic bounds due to Rao. We will also indicate a proof of Rao's inequalities for arbitrary asymmetric orthogonal arrays based on the same method.