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Also, we show that they are strongly isotopic if and only if $q\\equiv 1(mod\\,4)$. Consequently, for each $q\\equiv -1(mod\\,4)$ there exist isotopic commutative presemifields of order $q^{2\\ell}$ ($\\ell>1$ odd) defining CCZ--inequivalent planar DO polynomials."},"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":"1105.5940","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-30T10:49:58Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"2c21a616eadd7fa68acf82bffda2e2cb4cf530824ef723f5b1ab67fc4e0e60a8","abstract_canon_sha256":"8b38ca4ac2f827460b3842a7cf47d77440af04501af563567488df2e6ca68b57"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:07.578674Z","signature_b64":"JuDfgldjf3RKZbWZssiji2RcGMudEVO7ZIZ4vyyL7S/Q5ZxT9j2iIAOlswTjPEnRssEl/qrKlccrMNTemXfPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64202d1c641298c0e1b041dc85274b1a512e2e42e27cced730f29eccbf33c33b","last_reissued_at":"2026-05-18T04:14:07.578008Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:07.578008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On isotopisms and strong isotopisms of commutative presemifields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Giuseppe Marino, Olga Polverino","submitted_at":"2011-05-30T10:49:58Z","abstract_excerpt":"In this paper we prove that the $P(q,\\ell)$ ($q$ odd prime power and $\\ell>1$ odd) commutative semifields constructed by Bierbrauer in \\cite{BierbrauerSub} are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth in \\cite{BuHe2008}. Also, we show that they are strongly isotopic if and only if $q\\equiv 1(mod\\,4)$. Consequently, for each $q\\equiv -1(mod\\,4)$ there exist isotopic commutative presemifields of order $q^{2\\ell}$ ($\\ell>1$ odd) defining CCZ--inequivalent planar DO polynomials."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5940","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":"1105.5940","created_at":"2026-05-18T04:14:07.578113+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.5940v2","created_at":"2026-05-18T04:14:07.578113+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5940","created_at":"2026-05-18T04:14:07.578113+00:00"},{"alias_kind":"pith_short_12","alias_value":"MQQC2HDECKMM","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"MQQC2HDECKMMBYNQ","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"MQQC2HDE","created_at":"2026-05-18T12:26:34.985390+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/MQQC2HDECKMMBYNQIHOIKJ2LDJ","json":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ.json","graph_json":"https://pith.science/api/pith-number/MQQC2HDECKMMBYNQIHOIKJ2LDJ/graph.json","events_json":"https://pith.science/api/pith-number/MQQC2HDECKMMBYNQIHOIKJ2LDJ/events.json","paper":"https://pith.science/paper/MQQC2HDE"},"agent_actions":{"view_html":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ","download_json":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ.json","view_paper":"https://pith.science/paper/MQQC2HDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.5940&json=true","fetch_graph":"https://pith.science/api/pith-number/MQQC2HDECKMMBYNQIHOIKJ2LDJ/graph.json","fetch_events":"https://pith.science/api/pith-number/MQQC2HDECKMMBYNQIHOIKJ2LDJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ/action/storage_attestation","attest_author":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ/action/author_attestation","sign_citation":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ/action/citation_signature","submit_replication":"https://pith.science/pith/MQQC2HDECKMMBYNQIHOIKJ2LDJ/action/replication_record"}},"created_at":"2026-05-18T04:14:07.578113+00:00","updated_at":"2026-05-18T04:14:07.578113+00:00"}