Is there a binary [10,6,4] code?

Question

Using the sphere padding packing bound formula I can conclude that 1 + 12 + 66 $\ge$ $2^{6}$ which indicates that there MAY be a binary [10,6,4] code, however I cannot prove that there is. How can I come to the conclusion that this is or isn't a [10,6,4] code?

Answer 1

Hint(s):

1. If a linear $[n,k,d]$ code exists, than it is equivalent to a linear $[n,k,d]$ code having the generator matrix in the form of $G=[I_k|A]$, where $A$ is a $k * (n-k)$ matrix.
2. In a linear $[n,k,d]$ code, $d(C)=w(C)$. That means that if the minimum distance is $d$, than the minimum weight of any non-zero codeword is $d$ as well.

In case you are still confused, spoilered below are further hints/explanation:

Every codeword in the generator matrix is obviously a codeword in the linear code itself. In our [10,6,4] case, $G$ would be equivalent to $G=[I_6|A]$, where $A$ is a $6 * 4$ matrix. Every codeword in $G$ has a single $1$ in the first six positions coming from $I_6$, thus it must have at least three $1$s in the last 4 poisitions (must have minimum weight 4), all in $A$. Can you reach a contradiction from this?