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Hence, the extended quadratic residue codes hold an infinite family of $2$-designs for primes $n \\equiv 1 \\pmod{4}$, an infinite family of $3$-designs for primes $n \\equiv 3 \\pmod{4}$. To construct more $t$-designs with $t \\in \\{2, 3\\}$, one would search for other extended cyclic codes over finite fields that are invariant under"},"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":"1704.01199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-04-04T21:50:27Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"884d4cad10b52b413a1ed089f5be5d6e66888a83fa0650093220300bc007a034","abstract_canon_sha256":"d5eadf7f2df806e3afe800aa76627c93abf41dcdfa5ca275ed038b8552e144aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:58.817185Z","signature_b64":"Nxh+xLF6dzoRr8g5RHeAwqOBxhjYtQMtBLTcffUhhpEwuaf1CDreq4oY/k7KH3hL04YWvcwHq8T78RPeTvP4DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f52bd8ea4d65b4134cfbbdf2e2dd3f35f40443a6d88f0e9f9a5c5c20749bdfd3","last_reissued_at":"2026-05-18T00:46:58.816727Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:58.816727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"All binary linear codes that are invariant under $\\PSL_2(n)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Cunsheng Ding, Hao Liu, Vladimir D. 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To construct more $t$-designs with $t \\in \\{2, 3\\}$, one would search for other extended cyclic codes over finite fields that are invariant under"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.01199","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":"1704.01199","created_at":"2026-05-18T00:46:58.816800+00:00"},{"alias_kind":"arxiv_version","alias_value":"1704.01199v1","created_at":"2026-05-18T00:46:58.816800+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.01199","created_at":"2026-05-18T00:46:58.816800+00:00"},{"alias_kind":"pith_short_12","alias_value":"6UV5R2SNMW2B","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"6UV5R2SNMW2BGTH3","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"6UV5R2SN","created_at":"2026-05-18T12:31:03.183658+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/6UV5R2SNMW2BGTH3XXZOFXJ7GX","json":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX.json","graph_json":"https://pith.science/api/pith-number/6UV5R2SNMW2BGTH3XXZOFXJ7GX/graph.json","events_json":"https://pith.science/api/pith-number/6UV5R2SNMW2BGTH3XXZOFXJ7GX/events.json","paper":"https://pith.science/paper/6UV5R2SN"},"agent_actions":{"view_html":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX","download_json":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX.json","view_paper":"https://pith.science/paper/6UV5R2SN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1704.01199&json=true","fetch_graph":"https://pith.science/api/pith-number/6UV5R2SNMW2BGTH3XXZOFXJ7GX/graph.json","fetch_events":"https://pith.science/api/pith-number/6UV5R2SNMW2BGTH3XXZOFXJ7GX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX/action/storage_attestation","attest_author":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX/action/author_attestation","sign_citation":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX/action/citation_signature","submit_replication":"https://pith.science/pith/6UV5R2SNMW2BGTH3XXZOFXJ7GX/action/replication_record"}},"created_at":"2026-05-18T00:46:58.816800+00:00","updated_at":"2026-05-18T00:46:58.816800+00:00"}