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In this article, we propose new heuristics for minimising the amount of XOR gates required to compute a system of linear equations in GF(2). We first revisit the well known Boyar-Peralta strategy and argue that a proper randomisation process during the selection phases can lead to great improvements. We then propose new selection criteria and explain their rationale. Our new methods outperform state-of-the-art algorithms such as Paar or Boyar-Peralta (or open synthesis tools such as Yosys) whendoi:10.13154/tches.v2020.i1.203-230 dblp:journals/tches/TanP20 fatcat:sbt5zini6rcmjmdfjuijihaj24