{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:PEAGASLIB4ZFKKAKWNMKBZASNZ","short_pith_number":"pith:PEAGASLI","schema_version":"1.0","canonical_sha256":"79006049680f3255280ab358a0e4126e5d6ec96132c4d3892fc11393746a9b11","source":{"kind":"arxiv","id":"1208.5194","version":2},"attestation_state":"computed","paper":{"title":"On the Eigenvalues of Certain Matrices Over $\\mathbb{Z}_m$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Liang Feng Zhang","submitted_at":"2012-08-26T06:50:57Z","abstract_excerpt":"Let $m,n>1$ be integers and $\\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by elements of $\\mathbb{P}_{n,m}$, where $a_{uv}=1$ if the inner product $< u,v >=0$ and $a_{uv}=0$ otherwise. Let $B_{n,m}=A_{n,m}A_{n,m}^t$. The eigenvalues of $B_{n,m}$ have been studied by [1, 2, 3], where their applications in the study of expanders and locally decodable codes were described. In this paper, we completely determine the eigenvalues of $B_{n"},"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":"1208.5194","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-08-26T06:50:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"716ea644652d86967eb36da26d4ae254844133ac1af0e1945212359d66a02939","abstract_canon_sha256":"d5d8b17f792833a6ba3bcf97899d8416ac3c6c5cd5c376293bacd7380502e14f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:37.605016Z","signature_b64":"QexsqfpKaUPtze5FVrjoCXvGVrccKuMKOCxQkqlsLi5RskLXGR1TVkqdGWFpCn84INFwMxHPONDm3I+u3kElCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"79006049680f3255280ab358a0e4126e5d6ec96132c4d3892fc11393746a9b11","last_reissued_at":"2026-05-18T03:29:37.604351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:37.604351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Eigenvalues of Certain Matrices Over $\\mathbb{Z}_m$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Liang Feng Zhang","submitted_at":"2012-08-26T06:50:57Z","abstract_excerpt":"Let $m,n>1$ be integers and $\\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by elements of $\\mathbb{P}_{n,m}$, where $a_{uv}=1$ if the inner product $< u,v >=0$ and $a_{uv}=0$ otherwise. Let $B_{n,m}=A_{n,m}A_{n,m}^t$. The eigenvalues of $B_{n,m}$ have been studied by [1, 2, 3], where their applications in the study of expanders and locally decodable codes were described. In this paper, we completely determine the eigenvalues of $B_{n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5194","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":"1208.5194","created_at":"2026-05-18T03:29:37.604463+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.5194v2","created_at":"2026-05-18T03:29:37.604463+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5194","created_at":"2026-05-18T03:29:37.604463+00:00"},{"alias_kind":"pith_short_12","alias_value":"PEAGASLIB4ZF","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PEAGASLIB4ZFKKAK","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PEAGASLI","created_at":"2026-05-18T12:27:18.751474+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/PEAGASLIB4ZFKKAKWNMKBZASNZ","json":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ.json","graph_json":"https://pith.science/api/pith-number/PEAGASLIB4ZFKKAKWNMKBZASNZ/graph.json","events_json":"https://pith.science/api/pith-number/PEAGASLIB4ZFKKAKWNMKBZASNZ/events.json","paper":"https://pith.science/paper/PEAGASLI"},"agent_actions":{"view_html":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ","download_json":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ.json","view_paper":"https://pith.science/paper/PEAGASLI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.5194&json=true","fetch_graph":"https://pith.science/api/pith-number/PEAGASLIB4ZFKKAKWNMKBZASNZ/graph.json","fetch_events":"https://pith.science/api/pith-number/PEAGASLIB4ZFKKAKWNMKBZASNZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ/action/storage_attestation","attest_author":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ/action/author_attestation","sign_citation":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ/action/citation_signature","submit_replication":"https://pith.science/pith/PEAGASLIB4ZFKKAKWNMKBZASNZ/action/replication_record"}},"created_at":"2026-05-18T03:29:37.604463+00:00","updated_at":"2026-05-18T03:29:37.604463+00:00"}