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COD $\\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \\times n$ matrix, where nonzero entries are filled by $\\pm z_i$ or $\\pm z^*_i$, $i = 1, 2,..., k$, such that $\\mathcal{O}^H_z \\mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \\times n}$. Define $\\mathcal{O}_z$ a first type COD if and only if $\\mathcal{O}_z$ does not contain submatrix {\\pm z_j & 0; \\ 0 & \\pm z^*_j}$ or ${\\pm z^*_j & 0; \\ 0 & \\pm z_j}$. It is already known that, all CODs with maximal rate, i.e., maximal $k/p$, are of the first type.\n  In this pa"},"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":"1107.3268","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2011-07-17T01:18:46Z","cross_cats_sorted":["cs.DM","math.IT"],"title_canon_sha256":"44d2efa97049de64916f65121ecd1a7ca4e7f48e67321924a1ca7edd1486a3e8","abstract_canon_sha256":"43f7b5beffc29bca4b0b8b2f1d383a2856ff304ac8226527af5f75268a1ce6c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:51.351273Z","signature_b64":"0o1uK1aLPAf6e01xKGOxjEVsTmDBSq2CUKzZ6KXtDv4/7jCirMaSGgrPfu4HGYx5ZJbxP3Ha4ZLzqcr2zp6IAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b0e7643d81882a44bd60b780adddf13db2d8da8e7fe8662a8c4fda2ee4984d0","last_reissued_at":"2026-05-18T03:58:51.350786Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:51.350786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complex Orthogonal Designs with Forbidden $2 \\times 2$ Submatrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.IT"],"primary_cat":"cs.IT","authors_text":"Haibin Kan, Yuan Li","submitted_at":"2011-07-17T01:18:46Z","abstract_excerpt":"Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \\times n$ matrix, where nonzero entries are filled by $\\pm z_i$ or $\\pm z^*_i$, $i = 1, 2,..., k$, such that $\\mathcal{O}^H_z \\mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \\times n}$. Define $\\mathcal{O}_z$ a first type COD if and only if $\\mathcal{O}_z$ does not contain submatrix {\\pm z_j & 0; \\ 0 & \\pm z^*_j}$ or ${\\pm z^*_j & 0; \\ 0 & \\pm z_j}$. It is already known that, all CODs with maximal rate, i.e., maximal $k/p$, are of the first type.\n  In this pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3268","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":"1107.3268","created_at":"2026-05-18T03:58:51.350856+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.3268v2","created_at":"2026-05-18T03:58:51.350856+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3268","created_at":"2026-05-18T03:58:51.350856+00:00"},{"alias_kind":"pith_short_12","alias_value":"LMHHMQ6YDCBK","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"LMHHMQ6YDCBKIS6W","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"LMHHMQ6Y","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/LMHHMQ6YDCBKIS6WBN4AVXO7CP","json":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP.json","graph_json":"https://pith.science/api/pith-number/LMHHMQ6YDCBKIS6WBN4AVXO7CP/graph.json","events_json":"https://pith.science/api/pith-number/LMHHMQ6YDCBKIS6WBN4AVXO7CP/events.json","paper":"https://pith.science/paper/LMHHMQ6Y"},"agent_actions":{"view_html":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP","download_json":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP.json","view_paper":"https://pith.science/paper/LMHHMQ6Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.3268&json=true","fetch_graph":"https://pith.science/api/pith-number/LMHHMQ6YDCBKIS6WBN4AVXO7CP/graph.json","fetch_events":"https://pith.science/api/pith-number/LMHHMQ6YDCBKIS6WBN4AVXO7CP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP/action/storage_attestation","attest_author":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP/action/author_attestation","sign_citation":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP/action/citation_signature","submit_replication":"https://pith.science/pith/LMHHMQ6YDCBKIS6WBN4AVXO7CP/action/replication_record"}},"created_at":"2026-05-18T03:58:51.350856+00:00","updated_at":"2026-05-18T03:58:51.350856+00:00"}