{"paper":{"title":"Cyclic Projective Orbits on Rational Normal Curves and MDS Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pingzhi Yuan, Yangcheng Li","submitted_at":"2026-07-14T13:30:05Z","abstract_excerpt":"Let \\(A\\) be a cyclic operator on an \\(r\\)-dimensional vector space over a field \\(k\\), and let \\(z\\) be a cyclic vector. Their Krylov code has parity-check matrix \\((z,Az,\\ldots,A^{n-1}z)\\). For \\(r\\ge 3\\) and \\(n\\ge r+3\\), we prove that an MDS orbit segment lies on a rational normal curve precisely when the projective pair \\((A,[z])\\) is conjugate to one arising from the \\((r-1)\\)-st symmetric-power action of \\(\\mathrm{PGL}_2\\). Over finite fields, for companion operators, this gives a complete classification of the generalized Reed--Solomon locus into split semisimple, two nonsplit semisimp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12761","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/2607.12761/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"}