# Have Rackoff's coverability bounds been improved?

In "The covering and boundedness problems for vector addition systems", Rackoff considers a vector addition system $$(v,A)$$ of dimension $$k$$ and size $$n$$ and derives an upper bound of $$2^{2^{(\log_2 3)n(\log_2 n)}}$$ on the length of pointwise nonnegative (in author's terminology: "bounded") covering executions.

The paper is from 1978. Have stronger results been proven meanwhile in which, e.g., the constant $$\log_2 3$$ has been lowered?

Moreover, it seems to me that the same bound $$2^{2^{(\log_2 3)n(\log_2 n)}}$$ would be obtained using almost the same proof when re-interpreting $$n$$ as the maximal integer occurring in the addition set of the VAS and in the target to be covered. (As of now, $$n$$ is the size of the whole VAS.) Can anyone confirm or rejct this?