Lower bounds for \(q\)-ary codes of covering radius one (Q1567663)

Let \(k_q(n)\) denote the minimal cardinality of a \(q\)-ary code \(C\) of length \(n\) and covering radius one. In this paper the author derives lower bounds for \(k_q(n)\). As an application it is shown that the difference between \(k_q(n)\) and the sphere covering bound approaches infinity with increasing \(n\) if \(q\) is fixed and \((q-1)n+1\) does not divide \(q^n\). The author improves a few already known lower bounds listed in [\textit{G. Cohen}, \textit{I. Honkala}, \textit{S. Litsyn} and \textit{A. Lobstein}, Covering codes. North-Holland Mathematical Library. 54, Amsterdam: Elsevier (1997; Zbl 0874.94001)]. For instance, it is shown that \(k_2(21)\geq 95360\) and \(k_3(10)\geq 2835\).