  In this paper, we develop an orthogonal precoding scheme for integer-forcing
(IF) linear receivers using the steepest gradient algorithm. Although this
scheme can be viewed as a special case of the unitary precoded integer-forcing
(UPIF), it has two major advantages. First, the orthogonal precoding
outperforms its unitary counterpart in terms of achievable rate, outage
probability, and error rate. We verify this advantage via theoretical and
numerical analyses. Second, it exhibits lower complexity as the dimension of
orthogonal matrices is half that of unitary matrices in the real-valued domain.
For finding ``good'' orthogonal precoder matrices, we propose an efficient
algorithm based on the steepest gradient algorithm that exploits the
geometrical properties of orthogonal matrices as a Lie group. The proposed
algorithm has low complexity and can be easily applied to an arbitrary MIMO
configuration. We also confirm numerically that the proposed orthogonal
precoding outperforms UPIF type II in some scenarios and the X-precoder in
high-order QAM schemes, e.g., $64$- and $256$-QAM.
