This paper considers the design of linear transmit precoding schemes to minimize the weighted sum delay metric over a K-user multi-antenna multicast channel. Limited by the rank and power constraints, the precoding matrices are designed under two interesting scenarios. The first scenario assumes the availability of pilots that can be precoded, using which the transmitter can convey any choice of transmit precoders to the users. Consequently, the sought transmit precoders can be any complex-valued matrices subject to given rank (dimensionality) and power (norm) constraints. A provably convergent cyclic alternating ascent based algorithm is proposed for a relaxed version of the problem, and is shown to attain at least a stationary point. Assuming that no such pilots are available, the second scenario constrains the transmit precoders to lie in a finite codebook. A concatenation based approach is adopted for constructing higher rank precoding matrices, which can facilitate the precoder search and allow for efficient signaling. A simple deterministic algorithm is proposed which involves maximizing a submodular rate function per step, and yields a worst-case performance guarantee.