Recently, Guo and Xia gave sufficient conditions for an STBC to achieve full diversity when a PIC (Partial Interference Cancellation) or a PIC-SIC (PIC with Successive Interference Cancellation) decoder is used at the receiver. In this paper, we give alternative conditions for an STBC to achieve full diversity with PIC and PIC-SIC decoders, which are equivalent to Guo and Xia's conditions, but are much easier to check. Using these conditions, we construct a new class of full diversity PIC-SIC decodable codes, which contain the Toeplitz codes and a family of codes recently proposed by Zhang, Xu et. al. as proper subclasses. With the help of the new criteria, we also show that a class of PIC-SIC decodable codes recently proposed by Zhang, Shi et. al. can be decoded with much lower complexity than what is reported, without compromising on full diversity. © 2010 IEEE.