Reducibility of parameter ideals in low powers of the maximal ideal

A commutative noetherian local ring (R,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 l (depending on R) such that R is Gorenstein if and only if there exists an irreducible parameter ideal contained in ml. We give upper bounds for l that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.