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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)"},"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":"1903.07310","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-18T09:02:50Z","cross_cats_sorted":[],"title_canon_sha256":"6e0eb2a9a57b1e6e44a6af103d6842be3714a79a07ca872af174a9aeba4e8e07","abstract_canon_sha256":"25477f41fd8a21949dc656d3b907b6ef040bd63293342e1ad5205d002878298e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:01.775117Z","signature_b64":"3ZoeSblmzRSUQKRdLOvB+I2hNDHTij/YburVhqV1cXL7xDmpZGntYe4vvlHE3NXK69YOrQvA1M1tOeqJ2LwLBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"053eee1603106fd4bc61d3452ac1a974712c6f97ae543ec84f860407fe8ddd96","last_reissued_at":"2026-05-17T23:51:01.774692Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:01.774692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bilinear Forms on Finite Abelian Groups and Group-Invariant Butson Hadamard Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bernhard Schmidt, Tai Do Duc","submitted_at":"2019-03-18T09:02:50Z","abstract_excerpt":"Let $K$ be a finite abelian group and let $\\exp(K)$ denote the least common multiple of the orders of the elements of $K$. 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)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07310","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.07310","created_at":"2026-05-17T23:51:01.774759+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.07310v1","created_at":"2026-05-17T23:51:01.774759+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.07310","created_at":"2026-05-17T23:51:01.774759+00:00"},{"alias_kind":"pith_short_12","alias_value":"AU7O4FQDCBX5","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"AU7O4FQDCBX5JPDB","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"AU7O4FQD","created_at":"2026-05-18T12:33:12.712433+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR","json":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR.json","graph_json":"https://pith.science/api/pith-number/AU7O4FQDCBX5JPDB2NCSVQNJOR/graph.json","events_json":"https://pith.science/api/pith-number/AU7O4FQDCBX5JPDB2NCSVQNJOR/events.json","paper":"https://pith.science/paper/AU7O4FQD"},"agent_actions":{"view_html":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR","download_json":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR.json","view_paper":"https://pith.science/paper/AU7O4FQD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.07310&json=true","fetch_graph":"https://pith.science/api/pith-number/AU7O4FQDCBX5JPDB2NCSVQNJOR/graph.json","fetch_events":"https://pith.science/api/pith-number/AU7O4FQDCBX5JPDB2NCSVQNJOR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR/action/storage_attestation","attest_author":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR/action/author_attestation","sign_citation":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR/action/citation_signature","submit_replication":"https://pith.science/pith/AU7O4FQDCBX5JPDB2NCSVQNJOR/action/replication_record"}},"created_at":"2026-05-17T23:51:01.774759+00:00","updated_at":"2026-05-17T23:51:01.774759+00:00"}