  We are concerned with the numerical solution obtained by splitting methods of
certain parabolic partial differential equations. Splitting schemes of order
higher than two with real coefficients necessarily involve negative
coefficients. It has been demonstrated that this second-order barrier can be
overcome by using splitting methods with complex-valued coefficients (with
positive real parts). In this way, methods of orders 3 to 14 by using the
Suzuki--Yoshida triple (and quadruple) jump composition procedure have been
explicitly built. Here we reconsider this technique and show that it is
inherently bounded to order 14 and clearly sub-optimal with respect to error
constants. As an alternative, we solve directly the algebraic equations arising
from the order conditions and construct methods of orders 6 and 8 that are the
most accurate ones available at present time, even when low accuracies are
desired. We also show that, in the general case, 14 is not an order barrier for
splitting methods with complex coefficients with positive real part by building
explicitly a method of order 16 as a composition of methods of order 8.
