Several previous works have confirmed that a joint design that combines channel estimation, channel coding and space-time transmission can improve the system performanceover that of a separate design. These conclusions are however in general based on unstructured solutions obtained using computer search. The coding gain of these joint designs is thereforelimited by both the computer-searchable "short" code length and the compromise between "suboptimal" performance and "high" complexity of their optimal decoding.At this background, we propose a systematic space-time code construction for joint channel estimation and error correction for a two-transmit-antenna and half-rate system. Also proposed is itsmaximum- likelihood decoder that follows a priority-first search principle. Our systematic code construction, together with a fairly low-complexity optimal decoder, then allows one to work withlonger codes with no sacrifice in performance. For codes of short block length, our simulations illustrate that the codes we propose have comparable performance to the best computer-searchedcodes. For codes of long block lengths that are almost beyond the searchable range of existing computer systems, our codes are still better than some reference designs based on separatechannel estimation and error correction components.