{"id":"W4408734169","doi":"10.1007/s00373-025-02912-x","title":"Unbiased Hadamard Matrices and Ternary Self-dual Codes","year":2025,"lang":"en","type":"article","venue":"Graphs and Combinatorics","topic":"graph theory and CDMA systems","field":"Engineering","cited_by":1,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Lethbridge","funders":"Japan Society for the Promotion of Science; Natural Sciences and Engineering Research Council of Canada","keywords":"Mathematics; Hadamard transform; Ternary operation; Combinatorics; Dual (grammatical number); Complex Hadamard matrix; Discrete mathematics; Hadamard matrix; Computer science","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0006320812,0.0003973294,0.0004698137,0.002082563,0.0009090989,0.002224949,0.0004918062,0.001163692,0.003506728],"category_scores_gemma":[0.003594204,0.0003527791,0.0003171799,0.001349822,0.00158919,0.001904137,0.001048317,0.001043013,0.0004953747],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0007475783,"about_ca_system_score_gemma":0.000539274,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0007088539,"about_ca_topic_score_gemma":0.0007666394,"domain_scores_codex":[0.9993581,0.0002106784,0.00003112403,0.00009996152,0.0001916078,0.0001085618],"domain_scores_gemma":[0.9978412,0.001087695,0.0003685734,0.0002314406,0.0003391338,0.0001318613],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00002758836,0.00001163442,0.0001869004,0.00001767805,0.00000345691,0.00006259784,0.00009433621,0.00352081,0.001180809,0.9905642,0.0004454281,0.003884572],"study_design_scores_gemma":[0.00000714003,0.00000823771,0.0001514333,0.00001123334,0.000002945448,0.0001058752,0.00004462472,0.01853741,0.0009517082,0.9790668,0.001099465,0.00001305118],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.4879793,0.002843815,0.4141935,0.002640782,0.0004647685,0.00006447484,0.0004279265,0.00024621,0.09113923],"genre_scores_gemma":[0.9624084,0.0008258874,0.02135363,0.000375468,0.0002743689,0.00005198977,0.0001241853,0.00003703309,0.01454909],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.003506728,"threshold_uncertainty_score":0.01173115,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.003005978694838545,"score_gpt":0.177691623276308,"score_spread":0.1746856445814695,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}