Hi I'm trying to solve this problem but have some difficulty.
Write down a check matrix for the Hamming code of length $15$. How many code words are there? Assuming that the columns of your matrix are ordered in the natural order, which of the following are codewords?
$c_1 =011010110111000$
$c_2 = 00000000000011$
$c_3 =110110110111111$
I found a very similar question here at the forum: Parity Check Matrix From Hamming code length 15
From that I was pretty much handed the Check matrix (assuming it was right).
$H = \begin{bmatrix}0&0&0&0&0&0&0&1&1&1&1&1&1&1&1\\0&0&0&1&1&1&1&0&0&0&0&1&1&1&1&\\0&1&1&0&0&1&1&0&0&1&1&0&0&1&1\\1&0&1&0&1&0&1&0&1&0&1&0&1&0&1\end{bmatrix}$
My first question would be: why does this constitute a Check matrix?
Second I need to find the number of code words there are. I know that the solution is given by $2^k$ where $k$ is the dimension, but how do I get that from the check matrix?
Thirdly, in order to see if $c_i$ is a word I have the following to rely on: $s:=Hy^T$, $He^T=s$, decode by $y+e$ $(mod$ $2)$.
For the first word I get: $Hy^T=\begin{bmatrix}0&1&1&0\end{bmatrix}$, the sixth column in the check matrix. Thus $e=\begin{bmatrix}0&0&0&0&0&0&0&0&0&0&0&1&0&0&1\end{bmatrix}$. Now $e+y$ $(mod$ $2)$ is $\begin{bmatrix}0&1&1&0&1&0&1&1&0&1&1&0&0&0&1\end{bmatrix}$. Frome here I don't know how to proceed in order to show that $c_1$ is a code word. Help, please.