My task is to use syndromes to decode few words to their unique nearest neighbour
in C under IMLD.
I got my parity check matrix given which is equal
$ H = \begin{bmatrix}
X \\
I_{7}
\end{bmatrix} = \begin{bmatrix}
1& 1 & 1 & 1 & 0 & 0 & 0\\
0& 0 & 0 & 1 & 1 & 1 & 1\\
1& 1 & 0 &0 &0 &1 &1 \\
0& 1 & 1 &0 & 1 & 1 & 0\\
1&0 &0 &0 &0 &0 &0 \\
0& 1 & 0 & 0 & 0 & 0 &0 \\
0& 0 & 1 & 0 & 0 & 0 & 0\\
0& 0 & 0& 1 & 0 & 0 & 0\\
0& 0 & 0& 0 & 1 & 0 &0 \\
0& 0 & 0& 0 & 0 & 1 & 0\\
0& 0 & 0& 0 & 0 & 0 & 1
\end{bmatrix}$
To obtain the cosets and syndromes I need to get my Generator Matrix
Is it correct to get the Generator Matrix from Transformation matrix in the following steps:
1) Transpose H
2) Transofm it further to $RREF$
3) Remove the leading columns ?
Best Answer
You don't need the Generator Matrix to decode to any codeword. What you can do is the following. The syndrome vector $s=r*H$ where $r$ is a received $1\mathrm{X}N$ vector. The syndrome can be identified also as follows: Let the received vector be $r=c_{t}+e$ where $c_{t}$ is the transmitted codeword and $e$ is the error vector that corrupted the codeword. We all know that the Parity check matrix is the null space for the codewords. Now then $s = e*H$ and you have the position of the codeword which had an error in it. Say if $s=e*H$ picked up the 3rd column of $H^{T}$, then the 3rd bit was erreneous in the received vector $r$.