Hamming code matrix
Task number: 4036
Determine the generating matrix of the Hamming code \( H_3 \) (derived from the Fano plane).
Solution
The code has dimension 4, so it suffices to find 4 linearly independent vectors. If we take the codes corresponding to the lines passing through one point: 1010001, 1001010, 1100100, we find that they are linearly independent, because each of the other six vertices belongs to only one of these lines.
Their sum is the vector 1111111. With ithis vector we get the complement of any word.
It is necessary to find one vector that is linearly independent of the three, i.e. 0101001. Note that these vectors distinguish the remaining pairs of points from the initial three lines.
Answer
The matrix sought is, for example, the matrix \( \begin{pmatrix} 1&0&1&0&0&0&1\\ 1&0&0&1&0&1&0\\ 1&1&0&0&1&0&0\\ 0&1&0&1&0&0&1 \end{pmatrix} \).
