I am looking for the sharpest known upper bound on $K(n, 1)$ as $n \rightarrow \infty$. This is the minimal cardinality of a (not-necessarily linear) _covering code_ of $\{0, 1\}^n$ of radius 1.

In elementary terms: Using how few (possibly non-disjoint) Hamming balls of radius 1 can we cover $\{0, 1\}^n$? I am interested in upper-bounds to this quantity, especially asymptotically as $n \rightarrow \infty$. For example, even the statement $K(n, 1) \in o(2^n)$ I have not been able to find proven. (It's impossible to do strictly better than $\lceil{\frac{2^n}{n + 1}} \rceil$, by the sphere-covering bound.)

[As of 1998][1], _exact_ values of $K(n, 1)$ were only known for specific values of $n$. For example:

 - $K(2^k - 1, 1) = 2^{2^k - k - 1}$ (Hamming code)
 - $K(2^k, 1) = 2^{2^k - k}$ (Johnson 1972)
 - $K(2n + 1, 1) \leq 2^n \cdot K(n, 1)$ ([Cohen–Lobstein–Sloane 1986][2])
 - $K(K(n, 1) - 1, 1) \leq (n + 1) \cdot 2^{K(n, 1) - n - 1}$ (Cor. 1 of [Östergård and Kaikkonen][1])


  [1]: https://www.sciencedirect.com/science/article/pii/S0012365X97818250
  [2]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.392.3162