How to check whether a matrix is a parity check matrix for a Hamming Code Ham(2,5) or not:

Question

How to check whether a matrix is a parity check matrix for a Hamming Code Ham(2,5) or not:

The matrix is \begin{bmatrix} 4&1&4&2&0&4\\2&1&1&4&2&0\end{bmatrix}

Is there any criterion to check it?

Please help.I dont understand how is it done.Plesae give some help.

Answer 1

In this example, you need to check that each pair of columns is linearly independent as vectors over $\Bbb F_5$. For example, columns $1$ and $2$ are, since their determinant is $4\times1-1\times2\equiv2\pmod 5$. Over to you!