  We consider the three-dimensional $\pm J$ model defined on a simple cubic
lattice and study its behavior close to the multicritical Nishimori point where
the paramagnetic-ferromagnetic, the paramagnetic-glassy, and the
ferromagnetic-glassy transition lines meet in the T-p phase diagram (p
characterizes the disorder distribution and gives the fraction of ferromagnetic
bonds). For this purpose we perform Monte Carlo simulations on cubic lattices
of size $L\le 32$ and a finite-size scaling analysis of the numerical results.
The magnetic-glassy multicritical point is found at $p^*=0.76820(4)$, along the
Nishimori line given by $2p-1={\rm Tanh}(J/T)$. We determine the
renormalization-group dimensions of the operators that control the
renormalization-group flow close to the multicritical point, $y_1 = 1.02(5)$,
$y_2 = 0.61(2)$, and the susceptibility exponent $\eta = -0.114(3)$. The
temperature and crossover exponents are $\nu=1/y_2=1.64(5)$ and $\phi=y_1/y_2 =
1.67(10)$, respectively. We also investigate the model-A dynamics, obtaining
the dynamic critical exponent $z = 5.0(5)$.
