Mixing
Respresentation of Vectors spanned by basis vectors in a column space. Is the representation correct
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This question is related to coding theory. Consider a generator matrix $G$ of a systematic code. It is of the form $[I_k|P$. Any column in $P$ is a linear combination of some or all columns of$I_k$. Let $I_k$ be denoted by the set
$$ B={overline{m_1},dots,overline{m_k}}$$. Let any $overline{m_i}$ be a part of $n_i$ parity equations in $P$. These columns are denoted by the set
$$overline{m_i}_{(n_i)}$$
I will represent $P$, the parity columns of $G$ as
$$P = bigcup_{i=1}^{k} overline{m_i}_{(n_i)} $$.
Is this representation correct ?
Example: Let $G=[I_4|P]$ be given as
$$G=begin{array}{ccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 1 & 1 & 1 \ 0 & 0 & 1 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 1 & 0 & 1 & 1 end{array} $$
where P would be
$$P=begin{pmatrix} 1 & 0 & 1 \ 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 1 & 1 end{pmatrix}$$
let the three columns of $P$ be $p_1,p_2,p_3$ and
$p_1=overline{m_1}+overline{m_2}+overline{m_3}$
$p_2=overline{m_2}+overline{m_3}+overline{m_4}$
$p_3=overline{m_1}+overline{m_2}+overline{m_4}$.
Here, $n_1=2,n_2=3,n_3=2,n_4=2$. So,
$$P=overline{m_1}_{(2)} cup overline{m_2}_{(3)} cup overline{m_3}_{(2)} cup overline{m_4}_{(2)}$$
linear-algebra
$endgroup$
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$begingroup$
This question is related to coding theory. Consider a generator matrix $G$ of a systematic code. It is of the form $[I_k|P$. Any column in $P$ is a linear combination of some or all columns of$I_k$. Let $I_k$ be denoted by the set
$$ B={overline{m_1},dots,overline{m_k}}$$. Let any $overline{m_i}$ be a part of $n_i$ parity equations in $P$. These columns are denoted by the set
$$overline{m_i}_{(n_i)}$$
I will represent $P$, the parity columns of $G$ as
$$P = bigcup_{i=1}^{k} overline{m_i}_{(n_i)} $$.
Is this representation correct ?
Example: Let $G=[I_4|P]$ be given as
$$G=begin{array}{ccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 1 & 1 & 1 \ 0 & 0 & 1 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 1 & 0 & 1 & 1 end{array} $$
where P would be
$$P=begin{pmatrix} 1 & 0 & 1 \ 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 1 & 1 end{pmatrix}$$
let the three columns of $P$ be $p_1,p_2,p_3$ and
$p_1=overline{m_1}+overline{m_2}+overline{m_3}$
$p_2=overline{m_2}+overline{m_3}+overline{m_4}$
$p_3=overline{m_1}+overline{m_2}+overline{m_4}$.
Here, $n_1=2,n_2=3,n_3=2,n_4=2$. So,
$$P=overline{m_1}_{(2)} cup overline{m_2}_{(3)} cup overline{m_3}_{(2)} cup overline{m_4}_{(2)}$$
linear-algebra
$endgroup$
add a comment |
$begingroup$
This question is related to coding theory. Consider a generator matrix $G$ of a systematic code. It is of the form $[I_k|P$. Any column in $P$ is a linear combination of some or all columns of$I_k$. Let $I_k$ be denoted by the set
$$ B={overline{m_1},dots,overline{m_k}}$$. Let any $overline{m_i}$ be a part of $n_i$ parity equations in $P$. These columns are denoted by the set
$$overline{m_i}_{(n_i)}$$
I will represent $P$, the parity columns of $G$ as
$$P = bigcup_{i=1}^{k} overline{m_i}_{(n_i)} $$.
Is this representation correct ?
Example: Let $G=[I_4|P]$ be given as
$$G=begin{array}{ccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 1 & 1 & 1 \ 0 & 0 & 1 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 1 & 0 & 1 & 1 end{array} $$
where P would be
$$P=begin{pmatrix} 1 & 0 & 1 \ 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 1 & 1 end{pmatrix}$$
let the three columns of $P$ be $p_1,p_2,p_3$ and
$p_1=overline{m_1}+overline{m_2}+overline{m_3}$
$p_2=overline{m_2}+overline{m_3}+overline{m_4}$
$p_3=overline{m_1}+overline{m_2}+overline{m_4}$.
Here, $n_1=2,n_2=3,n_3=2,n_4=2$. So,
$$P=overline{m_1}_{(2)} cup overline{m_2}_{(3)} cup overline{m_3}_{(2)} cup overline{m_4}_{(2)}$$
linear-algebra
$endgroup$
This question is related to coding theory. Consider a generator matrix $G$ of a systematic code. It is of the form $[I_k|P$. Any column in $P$ is a linear combination of some or all columns of$I_k$. Let $I_k$ be denoted by the set
$$ B={overline{m_1},dots,overline{m_k}}$$. Let any $overline{m_i}$ be a part of $n_i$ parity equations in $P$. These columns are denoted by the set
$$overline{m_i}_{(n_i)}$$
I will represent $P$, the parity columns of $G$ as
$$P = bigcup_{i=1}^{k} overline{m_i}_{(n_i)} $$.
Is this representation correct ?
Example: Let $G=[I_4|P]$ be given as
$$G=begin{array}{ccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 1 & 1 & 1 \ 0 & 0 & 1 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 1 & 0 & 1 & 1 end{array} $$
where P would be
$$P=begin{pmatrix} 1 & 0 & 1 \ 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 1 & 1 end{pmatrix}$$
let the three columns of $P$ be $p_1,p_2,p_3$ and
$p_1=overline{m_1}+overline{m_2}+overline{m_3}$
$p_2=overline{m_2}+overline{m_3}+overline{m_4}$
$p_3=overline{m_1}+overline{m_2}+overline{m_4}$.
Here, $n_1=2,n_2=3,n_3=2,n_4=2$. So,
$$P=overline{m_1}_{(2)} cup overline{m_2}_{(3)} cup overline{m_3}_{(2)} cup overline{m_4}_{(2)}$$
linear-algebra
linear-algebra
edited Jan 18 at 12:33
asked Jan 18 at 12:10
user355370
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