Erasure codes have been widely employed in distributed storage systems to provide high data reliability at a cost of small redundancy. Modern distributed storage systems usually organize the storage nodes in racks, in which the cross-rack communication cost is much more expensive than the intra-rack communication cost. When the original data symbols stored in a single node are updated, it is critical to design erasure codes that can update the corresponding coded symbols with the cross-rack update bandwidth defined as the average amount of symbols transferred across different racks as small as possible. In this paper, we first derive a tight lower bound on the cross-rack update bandwidth under the condition of $(n,k)$ reconstruction property that is any $k$ out of the $n$ nodes can retrieve all the data symbols. Moreover, we derive the lower bound on redundancy subject to the minimum cross-rack update bandwidth. Furthermore, we propose explicit constructions that can achieve both the minimum cross-rack update bandwidth and the minimum redundancy.