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The rank of the communication matrix M_f for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree of f and D^CC(f) stand for the deterministic communication complexity for f(x\\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\\hat f||_1). In particular, the Log-rank conjecture holds for XOR "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.1245","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-04T04:59:41Z","cross_cats_sorted":[],"title_canon_sha256":"acb4a0b4f9cbb78e3caf1e0a415bef2096c6ff261fc19f1b9c77186ba45e3881","abstract_canon_sha256":"3f8c87a2fc3189256e557e2296b9ebbe526acfb3b9f652f297467fedd55693b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:32.700143Z","signature_b64":"aheNK/z8dg+58EnD9A+yDcxNxCtWMYW8o29fpiboAjE+YfiYhp1+MHtq14TXmPr1wsxzQavSR9IeXvZFGMejDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7860a912f5dfae894989a2fd427dfce407d9d1e7e4f661a59645570db925037","last_reissued_at":"2026-05-18T03:28:32.699460Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:32.699460Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fourier sparsity, spectral norm, and the Log-rank conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Chung Hoi Wong, Hing Yin Tsang, Ning Xie, Shengyu Zhang","submitted_at":"2013-04-04T04:59:41Z","abstract_excerpt":"We study Boolean functions with sparse Fourier coefficients or small spectral norm, and show their applications to the Log-rank Conjecture for XOR functions f(x\\oplus y) --- a fairly large class of functions including well studied ones such as Equality and Hamming Distance. The rank of the communication matrix M_f for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree of f and D^CC(f) stand for the deterministic communication complexity for f(x\\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\\hat f||_1). 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