{"paper":{"title":"Strong External Difference Families and Classification of $\\alpha$-valuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Donald L. Kreher, Douglas R. Stinson, Maura B. Paterson","submitted_at":"2024-06-13T12:58:47Z","abstract_excerpt":"One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\\mathbb{Z}_{a^2+1}$ makes use of $\\alpha$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a classification theorem which shows that all such $\\alpha$-valuations can be constructed recursively via a sequence of ``blow-up'' operations. We also enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\\mathbb{Z}_{a^2+1}$ for $a \\leq 14$ and we show that all these SEDFs are equivalent to $\\alpha$-valuations via affine transformations. Whether this holds for all $a > 14$ as "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.09075","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.09075/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}