{"paper":{"title":"Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Chang-An Zhao, Huachao Zhang","submitted_at":"2025-05-29T09:23:07Z","abstract_excerpt":"For a Kummer extension defined by the affine equation $y^{m}=\\prod_{i=1}^{r} (x-\\a_i)^{\\lambda_i}$ over\n  an algebraic extension $K$ of a finite field $\\fq$, where $\\la_i\\in \\Z\\backslash\\{0\\}$ for $1\\leq i\\leq r$, $\\gcd(m,q) = 1$, and $\\a_1,\\cdots,\\a_r\\in K$ are pairwise distinct elements,\n  we propose a simple and efficient method to find all pure gaps at many totally ramified places.\n  We introduce a bottom set of pure gaps and indicate that the set of pure gaps is completely determined by the bottom set.\n  Furthermore, we demonstrate that a pure gap can be deduced from a known pure gap by e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23274","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/2505.23274/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"}