TITLE: Is there a $(6,9,3)_2$ code?
QUESTION [12 upvotes]: 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?

REPLY [7 votes]: For a binary codeword of length $6$, there are $4$ choices for the last two coordinates: $00,01,10$ or $11$. Now assume that, there is a binary code with nine codewords. But then, at least three of the codewords must have the same the last two coordinates by pigeonhole principle. However, three codewords having the same last two coordinates cannot satisfy the given minimum distance since they must satisfy the same minimum distance for length $4$. Hence, there is no $(6,9,3)_2$ code.