We provide a simplified proof of the random k-XORSAT satisfiability threshold theorem. As an extension we also determine the full rank threshold for sparse random matrices over finite fields with precisely k non-zero entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao, Müller, Combinatorica, 2020]. The proof combines physics-inspired message passing arguments with a surgical moment computation.
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Coja‐Oghlan et al. (2024) studied this question.
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