Successive zero-forcing dirty paper coding (SZF-DPC) is a simplified alternative to DPC for MIMO broadcast channels (MIMO BCs). In the SZF-DPC scheme, the noncausally-known interference is canceled by DPC, while the residual interference is suppressed by the ZF technique. Due to the ZF constraints, the precoders are constrained to lie in the null space of a matrix. For the sum rate maximization problem under a sum power constraint, the existing precoder designs naturally rely on the singular value decomposition (SVD). The SVD-based design is optimal but needs high computational complexity. Herein, we propose two low-complexity optimal precoder designs for SZF-DPC, all based on the QR decomposition (QRD), which requires lower complexity than SVD. The first design method is an iterative algorithm to find an orthonormal basis of the null space of a matrix that has a recursive structure. The second proposed method, which will be shown to require the lowest complexity, results from applying a single QRD to the matrix comprising all users' channel matrices. We analytically and numerically show that the two proposed precoder designs are optimal.