  If a contact form on a (2n+1)-dimensional closed contact manifold admits
closed Reeb orbits, then its systolic ration is defined to be the quotient of
(n+1)th power of the shortest period of Reeb orbits by the contact volume. We
prove that every co-orientable contact structure on any closed contact manifold
admits a contact form with arbitrarily large systolic ratio. This statement
generalizes the recent result of Abbondandolo et al. in dimension three to
higher dimensions. We extend the plug construction of Abbondandolo et. al. to
any dimension, by means of generalizing the hamiltonian disc maps studied by
the authors to the symplectic ball of any dimension. The plug is a mapping
torus and it is equipped with a special contact form so that one can use it to
modify a given contact form if the Reeb flow leads to a circle bundle on a
"large" portion of the given contact manifold. Inserting the plug sucks up the
contact volume while the minimal period remains the same. Following the ideas
of Abbondandolo et al. and using Giroux's theory of Liouville open books, we
show that any co-orientable contact structure is defined by a contact form,
which is suitable to be modified via inserting plugs.
