{"paper":{"title":"All the stabilizer codes of distance 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"C.H. Oh, Juergen Bierbrauer, Qing Chen, Sixia Yu, Ying Dong","submitted_at":"2009-01-14T10:18:11Z","abstract_excerpt":"We give necessary and sufficient conditions for the existence of stabilizer codes $[[n,k,3]]$ of distance 3 for qubits: $n-k\\ge \\lceil\\log_2(3n+1)\\rceil+\\epsilon_n$ where $\\epsilon_n=1$ if $n=8\\frac{4^m-1}3+\\{\\pm1,2\\}$ or $n=\\frac{4^{m+2}-1}3-\\{1,2,3\\}$ for some integer $m\\ge1$ and $\\epsilon_n=0$ otherwise. Or equivalently, a code $[[n,n-r,3]]$ exists if and only if $n\\leq (4^r-1)/3, (4^r-1)/3-n\\notin\\lbrace 1,2,3\\rbrace$ for even $r$ and $n\\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n\\not=1$ for odd $r$. Given an arbitrary length $n$ we present an explicit construction for an optimal quantum stabiliz"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1968","kind":"arxiv","version":3},"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"}