Parity Check Matrix From Hamming code length 15

Question

I am not able to figure out whats the method the calculate the parity check matrix. I am not sure if this is the method which is 1 ,2 3,.........15. So I can write its binary 0001, 0010, 0011,..... 1111 and make it a 4X15 matrix will that be a parity check matrix

Answer 1

That is indeed a parity check matrix for a length 15 binary Hamming code.

In general, let $C$ be the code with parity check matrix

$$\begin{pmatrix} \mathbf{1}_2 & \mathbf{2}_2 & \cdots & \mathbf{r}_2 \end{pmatrix},$$

where $\mathbf{k}_2$ denotes the column vector containing the digits of $k$ in binary, padded with the appropriate number of $0$s at the top. Then $C$ is a length $2^r-1$ binary Hamming code.