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A $BH(K,h)$ matrix is a $K$-invariant $|K|\\times |K|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|K|I$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I$ is the identity matrix of order $|K|$. Let $\\nu_p(x)$ denote the $p$-adic valuation of the integer $x$. Using bilinear forms on $K$, we show that a $BH(K,h)$ exists whenever\n  (i) $\\nu_p(h) \\geq \\lceil \\nu_p(\\exp(K))/2 \\rceil$ for every prime divisor $p$ of $|K|$ and\n  (ii)","authors_text":"Bernhard Schmidt, Tai Do Duc","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-18T09:02:50Z","title":"Bilinear Forms on Finite Abelian Groups and Group-Invariant Butson Hadamard Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07310","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:636e7c8da77a694ec6ba1e66de89bc83d58f3a2c05c8f2c27e5adefcaaaad4b6","target":"record","created_at":"2026-05-17T23:51:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"25477f41fd8a21949dc656d3b907b6ef040bd63293342e1ad5205d002878298e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-18T09:02:50Z","title_canon_sha256":"6e0eb2a9a57b1e6e44a6af103d6842be3714a79a07ca872af174a9aeba4e8e07"},"schema_version":"1.0","source":{"id":"1903.07310","kind":"arxiv","version":1}},"canonical_sha256":"053eee1603106fd4bc61d3452ac1a974712c6f97ae543ec84f860407fe8ddd96","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"053eee1603106fd4bc61d3452ac1a974712c6f97ae543ec84f860407fe8ddd96","first_computed_at":"2026-05-17T23:51:01.774692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:01.774692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3ZoeSblmzRSUQKRdLOvB+I2hNDHTij/YburVhqV1cXL7xDmpZGntYe4vvlHE3NXK69YOrQvA1M1tOeqJ2LwLBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:01.775117Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.07310","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:636e7c8da77a694ec6ba1e66de89bc83d58f3a2c05c8f2c27e5adefcaaaad4b6","sha256:5aa56e1e26bc714b723906c7c5e9846762d7bbfa4d4995f71d503d3b51f170bb"],"state_sha256":"678a75b8ab8ce84db9f8d663fbb8515215cc50618c94da266faa1ed8ba423f55"}