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.
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