{"paper":{"title":"Systematic Constructions of Complementary Sets and Hadamard Matrices from Circulant Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Arbitrary Hadamard matrices act as seeds to generate Golay complementary sets and optimal cross-Z complementary sequence sets through circulant algebraic transformations.","cross_cats":[],"primary_cat":"eess.SP","authors_text":"Piyush Priyanshu, Subhabrata Paul, Sudhan Majhi","submitted_at":"2025-10-14T09:18:23Z","abstract_excerpt":"A Hadamard matrix $H$ of order $n$ is a square matrix with entries $\\pm 1$ satisfying $HH^T = nI_n$, where $I_n$ is the identity matrix of order $n$. A circulant Hadamard matrix is a Hadamard matrix whose rows are cyclic shifts of one another. This work establishes a unified algebraic framework that treats arbitrary Hadamard matrices as flexible seeds to systematically generate Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS) through algebraic transformations. In this paper, a systemat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This work establishes a unified algebraic framework that treats arbitrary Hadamard matrices as flexible seeds to systematically generate Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS) through algebraic transformations, providing the first generalized framework for constructing optimal CZCSS from arbitrary Hadamard seeds.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algebraic transformations applied to circulant Hadamard matrices of order 4 (or arbitrary order) preserve the required zero-correlation-zone properties for the resulting sequence sets at the claimed ratios (2/3 or 1/2) for arbitrary lengths.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A circulant-operator framework generates binary and complex complementary sequence sets with specified ZCZ ratios and optimal parameters from arbitrary Hadamard matrix seeds.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Arbitrary Hadamard matrices act as seeds to generate Golay complementary sets and optimal cross-Z complementary sequence sets through circulant algebraic transformations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6d309b8963d2026ef3f55ea99c60ba8204424635ac8dd13164f8059e0043b94e"},"source":{"id":"2510.12315","kind":"arxiv","version":3},"verdict":{"id":"c9a4723e-8bc8-43be-b063-d47e228a4e57","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T08:09:26.238345Z","strongest_claim":"This work establishes a unified algebraic framework that treats arbitrary Hadamard matrices as flexible seeds to systematically generate Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS) through algebraic transformations, providing the first generalized framework for constructing optimal CZCSS from arbitrary Hadamard seeds.","one_line_summary":"A circulant-operator framework generates binary and complex complementary sequence sets with specified ZCZ ratios and optimal parameters from arbitrary Hadamard matrix seeds.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The algebraic transformations applied to circulant Hadamard matrices of order 4 (or arbitrary order) preserve the required zero-correlation-zone properties for the resulting sequence sets at the claimed ratios (2/3 or 1/2) for arbitrary lengths.","pith_extraction_headline":"Arbitrary Hadamard matrices act as seeds to generate Golay complementary sets and optimal cross-Z complementary sequence sets through circulant algebraic transformations."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.12315/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":3,"snapshot_sha256":"3de1f534c775d8176b1efa9712deb35c34c102bc9a041fe5d1e6acb8c7849b88"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}