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The matrix $C(d)=R(d)-\\frac{1}{k}N(d)N(d)^\\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\\mu_1(d),...,\\mu_{v-1}(d)$. Let $\\D$ be the class of all binary designs of $\\D_{v,b,k}$. We prove that if there is a design $d^*\\i"},"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":"1205.5876","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-26T11:47:43Z","cross_cats_sorted":[],"title_canon_sha256":"0fa39564c7a8e8707af26d2df56666cc5d81757a833a86a3afbd7031a9fb7a1c","abstract_canon_sha256":"5cabe239fb309f206f147a19dcdaf281604080fb9c71e75ee1934eaffee07dcb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:06.065040Z","signature_b64":"9c4bh1CX5k98ysDRvRFxfD6N0WoR5D4kPYiovkeA/gVWjzaKeQn//o8BQKYCPhyRO1mQdwYN2qZb68HJqZFvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c30f47e3321b00ee15fee46736ae790e96e51f1c8d48cc3a67ff6f353b12701d","last_reissued_at":"2026-05-18T03:28:06.064211Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:06.064211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On optimality of designs with three distinct eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"E. Ghorbani, G. B. Khosrovshahi, M. R. Faghihi, S. Tat","submitted_at":"2012-05-26T11:47:43Z","abstract_excerpt":"Let $\\D_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\\in\\D_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix $C(d)=R(d)-\\frac{1}{k}N(d)N(d)^\\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\\mu_1(d),...,\\mu_{v-1}(d)$. Let $\\D$ be the class of all binary designs of $\\D_{v,b,k}$. We prove that if there is a design $d^*\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5876","kind":"arxiv","version":3},"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":"1205.5876","created_at":"2026-05-18T03:28:06.064329+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.5876v3","created_at":"2026-05-18T03:28:06.064329+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5876","created_at":"2026-05-18T03:28:06.064329+00:00"},{"alias_kind":"pith_short_12","alias_value":"YMHUPYZSDMAO","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"YMHUPYZSDMAO4FP6","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"YMHUPYZS","created_at":"2026-05-18T12:27:27.928770+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/YMHUPYZSDMAO4FP64RTTNLTZB2","json":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2.json","graph_json":"https://pith.science/api/pith-number/YMHUPYZSDMAO4FP64RTTNLTZB2/graph.json","events_json":"https://pith.science/api/pith-number/YMHUPYZSDMAO4FP64RTTNLTZB2/events.json","paper":"https://pith.science/paper/YMHUPYZS"},"agent_actions":{"view_html":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2","download_json":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2.json","view_paper":"https://pith.science/paper/YMHUPYZS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.5876&json=true","fetch_graph":"https://pith.science/api/pith-number/YMHUPYZSDMAO4FP64RTTNLTZB2/graph.json","fetch_events":"https://pith.science/api/pith-number/YMHUPYZSDMAO4FP64RTTNLTZB2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2/action/storage_attestation","attest_author":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2/action/author_attestation","sign_citation":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2/action/citation_signature","submit_replication":"https://pith.science/pith/YMHUPYZSDMAO4FP64RTTNLTZB2/action/replication_record"}},"created_at":"2026-05-18T03:28:06.064329+00:00","updated_at":"2026-05-18T03:28:06.064329+00:00"}