Reducibility of parameter ideals in low powers of the maximal ideal

A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer $\ell$ (depending on $R$) such that $R$ is Gorenstein if and only if there exists an irreducible parameter ideal contained in $\mathfrak{m}^\ell$. We give upper bounds for $\ell$ that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.