We show that no published march test detects all march-test detectable instances of linked faults in memories. We present necessary and sufficient conditions for detection of single cell linked faults. We identify the set of faults that are undetectable by march tests. We also present sets of faults that dominate all march-test detectable instances of linked multiple cell faults along with the necessary and sufficient conditions for their detection. Using a test generator that takes these conditions as input, we generate the first march tests that detect all march-test detectable linked faults. By considering the subsets of linked faults targeted by the well-known March A and March B tests, we also prove that these well-known tests are optimal for the corresponding sets of target faults.