Problem about constructing extended ternary golay code from ternary golay code
$begingroup$
I have one problem about some construction I'm about to explain.
I'm asked to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$. In my notes I'm using those generator matrices (respectively):
$G_{12}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 0 \
end{bmatrix}$
$G_{11}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 \
end{bmatrix}$
In the notes its says that $mathcal{G}_{12}$ is obtained from $mathcal{G}_{11}$ just adding whole 11 digits from a word of $mathcal{G}_{11}$ and then appending the result to the word. But I think it's wrong because you just have to take the first row of ${G}_{11}$ to show that it's not true.
So I have two questions:
- Wich rule I have to follow to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$?
- In the answer of the first question; how can I prove that for any word of $mathcal{G}_{11}$?
coding-theory
asked Dec 5 '18 at 19:11
LecterLecter
8110
$endgroup$
add a comment |
$begingroup$
I have one problem about some construction I'm about to explain.
I'm asked to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$. In my notes I'm using those generator matrices (respectively):
$G_{12}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 0 \
end{bmatrix}$
$G_{11}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 \
end{bmatrix}$
In the notes its says that $mathcal{G}_{12}$ is obtained from $mathcal{G}_{11}$ just adding whole 11 digits from a word of $mathcal{G}_{11}$ and then appending the result to the word. But I think it's wrong because you just have to take the first row of ${G}_{11}$ to show that it's not true.
So I have two questions:
- Wich rule I have to follow to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$?
- In the answer of the first question; how can I prove that for any word of $mathcal{G}_{11}$?
coding-theory
asked Dec 5 '18 at 19:11
LecterLecter
8110
$endgroup$
add a comment |
$begingroup$
I have one problem about some construction I'm about to explain.
I'm asked to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$. In my notes I'm using those generator matrices (respectively):
$G_{12}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 0 \
end{bmatrix}$
$G_{11}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 \
end{bmatrix}$
In the notes its says that $mathcal{G}_{12}$ is obtained from $mathcal{G}_{11}$ just adding whole 11 digits from a word of $mathcal{G}_{11}$ and then appending the result to the word. But I think it's wrong because you just have to take the first row of ${G}_{11}$ to show that it's not true.
So I have two questions:
- Wich rule I have to follow to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$?
- In the answer of the first question; how can I prove that for any word of $mathcal{G}_{11}$?
coding-theory
asked Dec 5 '18 at 19:11
LecterLecter
8110
$endgroup$
I have one problem about some construction I'm about to explain.
I'm asked to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$. In my notes I'm using those generator matrices (respectively):
$G_{12}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 0 \
end{bmatrix}$
$G_{11}=begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 \
0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 \
end{bmatrix}$
In the notes its says that $mathcal{G}_{12}$ is obtained from $mathcal{G}_{11}$ just adding whole 11 digits from a word of $mathcal{G}_{11}$ and then appending the result to the word. But I think it's wrong because you just have to take the first row of ${G}_{11}$ to show that it's not true.
So I have two questions:
- Wich rule I have to follow to construct $mathcal{G}_{12}$ from $mathcal{G}_{11}$?
- In the answer of the first question; how can I prove that for any word of $mathcal{G}_{11}$?
coding-theory
coding-theory
asked Dec 5 '18 at 19:11
LecterLecter
8110
asked Dec 5 '18 at 19:11
LecterLecter
8110
edited Dec 5 '18 at 19:22
Lecter
asked Dec 5 '18 at 19:11
LecterLecter
8110
asked Dec 5 '18 at 19:11
LecterLecter
8110
asked Dec 5 '18 at 19:11
LecterLecter
8110
8110
add a comment |
add a comment |
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