We prove that the smallest covering code of length \(8\) and covering radius \(2\) has exactly \(12\) words. The proof is based on partial classification of even weight codewords, followed by a search for small sets of odd codewords covering the part of the space that has not been covered by the even subcode.
Citation
Uri Blass, Simon Litsyn. The Smallest Covering Code of Length \(8\) and Radius \(2\) has \(12\) Words[J], Ars Combinatoria, Volume 052. 309-318. .