Does there exist a $1$-error-correcting binary code with block length $6$ and $9$ codewords?
The Hamming bound says that for any code $C$ with those parameters, $|C| \le \frac{2^6}{1+6} \approx 9.14$. So, we can't rule out the existence of such a code using the Hamming bound.
The Singleton bound says that for any code $C$ with those parameters, $|C| \le 2^{6-3+1} = 16$, so we can't rule out the existence of such a code using the Singleton bound.
Also, this can't be a linear code, since $\log_2 9$ isn't an integer. That rules out the possibility of just enumerating the possible generator matrices.
I feel a bit silly asking this, but the direct approach (assume $000\ 000$ is in the code, then all others must have Hamming weight $\ge 3$, ...) becomes unmanageable quickly. How else can I go about this?