{"paper":{"title":"Linearity and Complements in Projective Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Alexander Vardy, Michael Braun, Tuvi Etzion","submitted_at":"2011-03-16T07:48:11Z","abstract_excerpt":"The projective space of order $n$ over the finite field $\\Fq$, denoted here as $\\Ps$, is the set of all subspaces of the vector space $\\Fqn$. The projective space can be endowed with distance function $d_S(X,Y) = \\dim(X) + \\dim(Y) - 2\\dim(X\\cap Y)$ which turns $\\Ps$ into a metric space. With this, \\emph{an $(n,M,d)$ code $\\C$ in projective space} is a subset of $\\Ps$ of size $M$ such that the distance between any two codewords (subspaces) is at least $d$. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an $(n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3117","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"}