Combinatorial lower bounds on binary codes with covering radius one (Q2713644)

Consider binary codes of covering radius 1 and length \(n\), and denote by \(K(n,1)\) the size of the smallest of them. The paper starts by giving a short survey of lower estimates for \(K(n,1)\), and then reports two kinds of improvements described by \textit{L.~Habsieger} in [Discrete Math. 176, 115-130 (1997; Zbl 0898.94016)]. By a sharper estimate of certain sets, the author gets an improvement for the case \(n \equiv 5\bmod 6\), and also for cases \(n \equiv 19 \bmod 60\) and \(n \equiv 39 \bmod 60\).