{"paper":{"title":"A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"q-ary NMDS codes that support 2-designs can be built generically with lengths exceeding q+1","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Chunming Tang, Dongchun Han, Hao Chen, Hengfeng Liu, Zhengchun Zhou","submitted_at":"2025-06-20T07:17:17Z","abstract_excerpt":"A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial $t$-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for $t > 2$, all with length at most $q + 1$. Although NMDS codes can, in princ"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present the first generic construction of q-ary NMDS codes supporting 2-designs with lengths exceeding q + 1, resulting in an infinite family of such codes along with their weight distributions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The claimed new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs actually produce NMDS codes that support 2-designs for lengths beyond q+1 (abstract, paragraph on method).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A generic construction produces an infinite family of q-ary NMDS codes supporting 2-designs with lengths exceeding q+1 and explicit weight distributions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"q-ary NMDS codes that support 2-designs can be built generically with lengths exceeding q+1","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8901c4fd961f70465e85008ad9e457dc10d0352870fd2412c3093ce4d485762a"},"source":{"id":"2506.16793","kind":"arxiv","version":3},"verdict":{"id":"361f0509-d938-42b6-a508-23b5e073c62f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T08:33:31.474895Z","strongest_claim":"We present the first generic construction of q-ary NMDS codes supporting 2-designs with lengths exceeding q + 1, resulting in an infinite family of such codes along with their weight distributions.","one_line_summary":"A generic construction produces an infinite family of q-ary NMDS codes supporting 2-designs with lengths exceeding q+1 and explicit weight distributions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The claimed new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs actually produce NMDS codes that support 2-designs for lengths beyond q+1 (abstract, paragraph on method).","pith_extraction_headline":"q-ary NMDS codes that support 2-designs can be built generically with lengths exceeding q+1"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.16793/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":62,"sample":[{"doi":"","year":1969,"title":"E.F. Assmus Jr., H.F. Mattson Jr., New 5-designs, J. Combinat. Theory 6 (2) (1969) 122–151. 26","work_id":"9342e09c-56de-4f65-af94-175a6bec1d78","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"T. Beth, D. Jungnickel, H. Lenz, Design Theory, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1999","work_id":"0bdc8d77-7dee-42e6-bc81-41490df5fddb","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"De Boer, Almost MDS codes, Des","work_id":"520defbd-79fc-40ec-abcd-b8a421cc05a5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Colbourn, CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL","work_id":"9eead0c1-87c0-405f-9807-2cb7941f7ff1","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Ding, Infinite families of 3-designs from a type of five-weight code, Des","work_id":"eb5e1b18-2fea-41a4-a8a2-70cf0d1be58a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":62,"snapshot_sha256":"6b277eba8b2162fdb03e7b2ec2de60a5c643ba51c4312d6e8c4e8bbf667dc5f0","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"}