Combinatorial Lower Bounds on Binary Codes with Covering Radius One

We give a short survey of the best known lower bounds on \(K(n, 1)\), the minimum cardinality of a binary code of length \(n\) and covering radius \(1\). Then we prove new lower bounds on \(K(n, 1)\), e.g.
\[K(n,1)\geq \frac{(5n^2-13n+66)2^n}{(5n^2-13n+46)(n+1)}\] when \(n \equiv 5 \pmod{6}\)
which lead to several numerical improvements.