  Lattice precoding is an effective strategy for multiantenna broadcast. In
this paper, we show that approximate lattice precoding in multiantenna
broadcast is a variant of the closest vector problem (CVP) known as $\eta$-CVP.
The proximity factors of lattice reduction-aided precoding are defined, and
their bounds are derived, which measure the worst-case loss in power efficiency
compared to sphere precoding. Unlike decoding applications, this analysis does
not suffer from the boundary effect of a finite constellation, since the
underlying lattice in multiantenna broadcast is indeed infinite.
