In this paper mD convolutional codes with finite support are considered, which are convolutional codes whose codewords have compact support indexed in \(\mathbb {N}^{m}\) and take values in \(\mathbb {F}^{n}\) , where \(\mathbb {F}\) is a finite field. We establish a natural upper bound on the distance of an mD convolutional code of rate k/n and degree \(\delta \) , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. We then present a particular construction of 3D convolutional codes with finite support of rate 1/n and degree \(\delta \le 2\) that achieve such bound and hence have the maximum distance among all 3D convolutional codes with the same rate and degree.