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Consider the binary linear $(13, 64, 5)$-code $C_3$, or a $[13, 6, 5]$-code.

Then, let $M$ be the generator matrix for $C_3$, such that we can rearrange the coordinate positions of the words in $C_3$ so that $M$ is in the form:

$$\left[\begin{array} {ccccc|cccccccc} 1&1&1&1&1&0&0&0&0&0&0&0&0 \\ &&G_1 &&&&&&&G_2 \\ \end{array}\right]$$

I have to show that $G_2$ is the generator matrix for a linear $[8, 5, 3]$ code. It is clear that if $G_2$ is a generator matrix, then the length of the code is $n=8$ and that the dimension is $k=6-1=5$. It is also clear that if $G_2$ is a generator for this code, then the code is linear by definition.

However, how do I show that $G_2$ is in fact a generator matrix? I was thinking of using the fact that a generator matrix has linearly independent rows. But just because $M$ has linearly indepenedet rows does not mean that $G_2$ does as well. So I am not sure why we can say that $G_2$ is a generator matrix.

I am also stuck on showing that the minimum distance would be $3$. I was thinking of using that the code $C_3$ has minimum distance $5$, but I am not sure how.

Any help is appreciated, thank you!

Edit: $C_3$ is obtained by assuming that there exists a linear $(16, 256, 6)$ code for contradiction, then removing the last digit to get a linear $(15, 256, 5)$ code, then cross section $x_1=0$ to get a linear $(14, 128, 4)$ code, and then cross section $x_1=0$ again to get $C_3$, a linear $(13, 64, 5)$ code.

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From your details, your code is binary (you should mention that).

You always have that $G_{2}$ generates some code (its row space). To show that it is a generator matrix for that code, you need to show that it has full row rank.

You know that $\begin{bmatrix} G_{1} & G_{2} \end{bmatrix}$ has full row rank 5. To show that $G_{2}$ also has full row rank, consider starting with $M$ in standard form, that is $M = \begin{bmatrix} I_{6} & A \end{bmatrix}$. When you reorder the columns to put $M$ into the form you have given, what happened to the other 5 pivots that made up the $I_{6}$?

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In fact, using Hamming bound, we know $A_2(8,3) \le \frac{2^8}{1+8} <2^5$, so there is no $[8,5,3]$ binary code. Consequently, there is no point discussing the generator matrix of a non-existent code.

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    $\begingroup$ Of course there is a point. If you show that $G_{2}$ is the generator matrix for a binary $[8,5,3]$ code, then the nonexistence of an $[8,5,3]$ code gives you the nonexistence of a binary $[13,6,5]$ code (which is not ruled out by the Hamming bound). $\endgroup$ May 15, 2018 at 4:36

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