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By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $v\\in \\Phi_1\\cup \\Phi_2 \\cup \\Phi_3 \\cup \\Phi_4$; and for $u\\in \\{3,5\\}$ and $v\\in \\Phi_1\\cup \\Phi_2 \\cup \\Phi_3$. Here, $\\Phi_1=\\{q^2:q\\equiv 1\\pmod{4}\\mbox{ is a prime power}\\}$, $\\Phi_2=\\{n^4\\in \\mathbb{N}:n\\equiv 1\\pmod{2}\\} \\cup \\{9n^4\\in \\mat"},"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":"1809.05253","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-14T04:53:24Z","cross_cats_sorted":[],"title_canon_sha256":"40b5d2fb03403118d6df30ccb28fc51f5c1952a94ae8fc1d176f755b8d6c9802","abstract_canon_sha256":"3eca964636e0d69ba39387f0e03b3360b9e94f871429bfefc2b289511df89927"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:43.666028Z","signature_b64":"i33dZzASf15rByec8tM3Zn+4jMrH/LisN+MsIJG/UTiUPGaCm1G9KYVrnZGL6SJhlCVy6h3BdC2jesH0QrU/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92d417249f221670c7caebf16c0ee9e735104e11584691a25ef9752a044d6dc3","last_reissued_at":"2026-05-17T23:40:43.665420Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:43.665420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New constructions of Hadamard matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ka Hin Leung, Koji Momihara","submitted_at":"2018-09-14T04:53:24Z","abstract_excerpt":"In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $v\\in \\Phi_1\\cup \\Phi_2 \\cup \\Phi_3 \\cup \\Phi_4$; and for $u\\in \\{3,5\\}$ and $v\\in \\Phi_1\\cup \\Phi_2 \\cup \\Phi_3$. Here, $\\Phi_1=\\{q^2:q\\equiv 1\\pmod{4}\\mbox{ is a prime power}\\}$, $\\Phi_2=\\{n^4\\in \\mathbb{N}:n\\equiv 1\\pmod{2}\\} \\cup \\{9n^4\\in \\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05253","kind":"arxiv","version":2},"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":"1809.05253","created_at":"2026-05-17T23:40:43.665511+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.05253v2","created_at":"2026-05-17T23:40:43.665511+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.05253","created_at":"2026-05-17T23:40:43.665511+00:00"},{"alias_kind":"pith_short_12","alias_value":"SLKBOJE7EILH","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"SLKBOJE7EILHBR6K","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"SLKBOJE7","created_at":"2026-05-18T12:32:53.628368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1907.02623","citing_title":"A new family of Hadamard matrices of order $4(2q^2+1)$","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44","json":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44.json","graph_json":"https://pith.science/api/pith-number/SLKBOJE7EILHBR6K5PYWYDXJ44/graph.json","events_json":"https://pith.science/api/pith-number/SLKBOJE7EILHBR6K5PYWYDXJ44/events.json","paper":"https://pith.science/paper/SLKBOJE7"},"agent_actions":{"view_html":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44","download_json":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44.json","view_paper":"https://pith.science/paper/SLKBOJE7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.05253&json=true","fetch_graph":"https://pith.science/api/pith-number/SLKBOJE7EILHBR6K5PYWYDXJ44/graph.json","fetch_events":"https://pith.science/api/pith-number/SLKBOJE7EILHBR6K5PYWYDXJ44/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44/action/storage_attestation","attest_author":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44/action/author_attestation","sign_citation":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44/action/citation_signature","submit_replication":"https://pith.science/pith/SLKBOJE7EILHBR6K5PYWYDXJ44/action/replication_record"}},"created_at":"2026-05-17T23:40:43.665511+00:00","updated_at":"2026-05-17T23:40:43.665511+00:00"}