  We study absorbing phase transitions in the one-dimensional branching
annihilating random walk with long-range repulsion. The repulsion is
implemented as hopping bias in such a way that a particle is more likely to hop
away from its closest particle. The bias strength due to long-range interaction
has the form $\varepsilon x^{-\sigma}$, where $x$ is the distance from a
particle to its closest particle, $0\le \sigma \le 1$, and the sign of
$\varepsilon$ determines whether the interaction is repulsive (positive
$\varepsilon$) or attractive (negative $\varepsilon$). A state without
particles is the absorbing state. We find a threshold $\varepsilon_s$ such that
the absorbing state is dynamically stable for small branching rate $q$ if
$\varepsilon < \varepsilon_s$. The threshold differs significantly, depending
on parity of the number $\ell$ of offspring. When $\varepsilon>\varepsilon_s$,
the system with odd $\ell$ can exhibit reentrant phase transitions from the
active phase with nonzero steady-state density to the absorbing phase, and back
to the active phase. On the other hand, the system with even $\ell$ is in the
active phase for nonzero $q$ if $\varepsilon>\varepsilon_s$. Still, there are
reentrant phase transitions for $\ell=2$. Unlike the case of odd $\ell$,
however, the reentrant phase transitions can occur only for $\sigma=1$ and
$0<\varepsilon < \varepsilon_s$. We also study the crossover behavior for $\ell
= 2$ when the interaction is attractive (negative $\varepsilon$), to find the
crossover exponent $\phi=1.123(13)$ for $\sigma=0$.
