How to find actual code based on parity check matrix, generator matrix and received code?
We are given the parity check equations:$$\begin{align} x_5 &= x_1~x_3~x_4\\ x_6 &= x_1~x_2~x_3\\ x_7 &= x_2~x_3~x_4 \end{align}$$the generator matrix, $G$ is$$\begin{align} 1000&~110\\ 0100&~011\\ 0010&~111\\ 0001&~101 \end{align}$$The parity check matrix, $H$ is:$$\begin{align} 1011&~100\\ 1110&~010\\ 0111&~001 \end{align}$$we are given $x' = 1010~100$ as the recieved message. It goes from $Z(4,2)$ to $Z(7,2)$, so the actual code word is $4$ characters long. I need to find the actual code word. I did $x'(H)$ and got $[101]$.
At this point, my notes say $101$ is the $4$th column of $H$, and that the corrected code is $1011~100$.
So what is the actual code that was sent in this case?
(Not sure if any additional info is needed to answer this but I can check for more info if needed).
$\endgroup$ 61 Answer
$\begingroup$This is a systematic code with generator matrix in the form $[I|A]$ thus the first 4 bits of the received codeword make up the actual data word that was sent. Thus the data word sent was $[1011]$.
In another word the codeword is [clearly from the matrix, and the equations]
$$[x_1, x_2, x_3, x_4, x_1+x_3+x_4,x_1+x_2+x_3,x_2+x_3+x_4]$$
$\endgroup$
