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We prove a lower bound on the communication complexity of computing the n-fold xor of an arbitrary function f, in terms of the communication complexity and rank of f. We prove that D(f⊕ n) ≥ n · (Ω(D(f))/log rk(f) -log rk(f) ), where here D(f), D(f⊕ n) represent the deterministic communication complexity, and rk(f) is the rank of f. Our methods involve a new way to use information theory to reason about deterministic communication complexity.
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Iyer et al. (2024) studied this question.
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