Vote assignments are being used for the design of fault tolerant systems. They work in such a way that every node in a computer network gets assigned a certain number of votes. Furthermore, a quorum is defined. A transaction can only take place if the number of votes collected is greater than or equal to the quorum. These single dimensional vote assignments provide a compact representation of certain antichains, namely those which consist of the minimal sets of nodes achieving the quorum. Multidimensional (MD) vote assignments are a generalization of single dimensional vote assignments. In contrast to single dimensional vote assignments, MD vote assignments have the important property that every antichain has a corresponding MD vote assignment and vice versa. For the efficient design of fault tolerant systems, it is important that the dimension of a MD vote assignment is as small as possible. We introduce some new heuristics for calculating a MD vote assignment which perform better than the heuristics known so far.