  Low Earth orbit (LEO) satellite constellations rely on inter-satellite links
(ISLs) to provide global connectivity. However, one significant challenge is to
establish and maintain inter-plane ISLs, which support communication between
different orbital planes. This is due to the fast movement of the
infrastructure and to the limited computation and communication capabilities on
the satellites. In this paper, we make use of antenna arrays with either Butler
matrix beam switching networks or digital beam steering to establish the
inter-plane ISLs in a LEO satellite constellation. Furthermore, we present a
greedy matching algorithm to establish inter-plane ISLs with the objective of
maximizing the sum of rates. This is achieved by sequentially selecting the
pairs, switching or pointing the beams and, finally, setting the data rates.
Our results show that, by selecting an update period of 30 seconds for the
matching, reliable communication can be achieved throughout the constellation,
where the impact of interference in the rates is less than 0.7 % when compared
to orthogonal links, even for relatively small antenna arrays. Furthermore,
doubling the number of antenna elements increases the rates by around one order
of magnitude.
