{"paper":{"title":"On lifting perfect codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Josep Rif\\`a, Victor Zinoviev","submitted_at":"2010-02-01T17:54:36Z","abstract_excerpt":"In this paper we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given perfect code C of length n=(q^m-1)/(q-1) over F_q with a parity check matrix H_m, we define a new code C_{(m,r)} of length n over F_{q^r}, r > 1, with this parity check matrix H_m. The resulting code C_{(m,r)} is completely regular with covering radius R = min{r,m}. We compute the intersection numbers of such codes and, finally, we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.0295","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":""},"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"}