LT (Luby transform) codes are a celebrated family of rateless erasure codes (RECs). Most of existing LT codes were designed for applications in which a centralized encoder possesses all message blocks and is solely responsible for encoding them into codewords. Distributed LT codes, in which message blocks are physically scattered across multiple different locations (encoders) that need to collaboratively perform the encoding, has never been systemically studied before despite its growing importance in applications. In this work, we present the first systemic study of LT codes in the distributed setting, and make the following three major contributions. First, we show that only a proper subset of LT codes are feasible in the distributed setting, and give the sufficient and necessary condition for such feasibility. Second, we propose a distributed encoding protocol that can efficiently implement any feasible code. The protocol is parameterized by a so-called action probability array (APA) that is only a few KBs in size, and any feasible code corresponds to a valid APA setting and vice versa. Third, we propose two heuristic search algorithms that have led to the discovery of feasible codes that are much more efficient than the state of the art.