# Is Rackoff's bound on the length of shortest covering executions in VAS still the best known one?

In 1976–1977, Rackoff proved in The covering and boundedness problems for vector addition systems that the length of a shortest covering sequence in a VAS is bounded by $$2^{(3n)^n} = 2^{2^{(\log_2 3)n\log_2 n}}$$, where $$n$$ is the input size. Is this upper bound still the best known one in general or has it been tightened (e.g., to something like $$2^{(dn)^n}$$ for $$d<3$$) in the meantime? I searched for improvements in the literature but have not discovered anything.