{"paper":{"title":"New Identities Relating Wild Goppa Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.NT"],"primary_cat":"cs.IT","authors_text":"Alain Couvreur, Ayoub Otmani, Jean-Pierre Tillich","submitted_at":"2013-10-11T17:17:55Z","abstract_excerpt":"For a given support $L \\in \\mathbb{F}_{q^m}^n$ and a polynomial $g\\in \\mathbb{F}_{q^m}[x]$ with no roots in $\\mathbb{F}_{q^m}$, we prove equality between the $q$-ary Goppa codes $\\Gamma_q(L,N(g)) = \\Gamma_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\\cdots +q+1}.$ In particular, for $m=2$, that is, for a quadratic extension, we get $\\Gamma_q(L,g^q) = \\Gamma_q(L,g^{q+1})$. If $g$ has roots in $\\mathbb{F}_{q^m}$, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.3202","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}