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How can i create a generator matrix for $(5,3)$ Linear block-code over $GF(2^2)$.

Most of the books mention the operations on this matrix and also that the choice of the basis vector is not unique that is the reason that a code can have different generator matrices. I know the rows of the matrix should be linearly independen but i don't understand how do i create generator matrix over a Galois field?

A concrete question because the above written question was not clear:

Suppose i have a paritytestmatrix = $\begin{bmatrix} 1 & \alpha & \alpha^2 & 1 & 0 \\ 1 &1 & 1& 0 &1 \\ \end{bmatrix}$

$\alpha$ is a root of $x^2 + x + 1$. Now i know that $G = [I_k|P]$. How do i go in opposite direction? Inverse of a indentity matrix is same.

A little bit explaination would be enough.

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  • $\begingroup$ Your question is not clear. There are many $(8,3)$ codes. You mean you are given a particular code and you want to find its $G$ (generator matrix)? Or just find some arbritrary $G$ ? In the latter case, just find any $3$ linearly indepedent binary tuples (rows, vectors)of length 8. Or you mean you don't understand when binary tuples are LI? $\endgroup$ – leonbloy Jan 26 '19 at 1:48
  • $\begingroup$ @leonbloy i have added a concrete example. If you can explain, it would be great help. $\endgroup$ – Khan Saab Jan 26 '19 at 2:18
  • $\begingroup$ Sorry, I had read $GF(2)$ :-) $\endgroup$ – leonbloy Jan 26 '19 at 3:45
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So you mean, how to find $G$ from a given $H$.

Recall that in general, if $G = [I_k|P]$ then $H=[-P^t | I_{n-k}]$ is a valid parity check matrix, because $G H^t=0$ (hence each row of $H$ is orthogonal to each row of $G$). And viceversa.

If the given matrix ($G$ or $H$) is not writen in the systematic way, then, we resort to the standard recipe of doing elementary row operations to put in that way (if this is not enough, we must permute columns and repermute them afterwards).

But here $H$ is systematic. So you just write

$$ -P^t= \begin{bmatrix} 1 & \alpha & \alpha^2 \\ 1 &1 & 1 \\ \end{bmatrix}$$

from that you get $P$ (remember that negation is performed in $GF(2^2)$ here) and you get $G$.

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