Check a word given a generator matrix G

Question

Take the binary code C with generator matrix: $$ \left( \begin{array}{ccccc|cccc} 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ \end{array} \right) $$ And use it to correct the word that has at most one error: $011000011$.

Notice that this matrix is of the form $(I_5 | P)$, we obtain the parity matrix by transposition: $(P^T|I_{9-5})=(P^T|I_{4})$, this gives us the following parity matrix: $$ \left( \begin{array}{ccccc|cccc} 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ We obtain the four equations which each codeword must satisfy, where $m_i$ denotes the message bits and $r_i$ denotes the parity bit: $$m_1 + m_3 +m_4+m_5 + r_1 = 0$$ $$m_1 + m_2 +m_4+m_5 + r_2 = 0$$ $$m_1 + m_2 +m_3+m_5 + r_3 = 0$$ $$m_2 + m_3 +m_4+m_5 + r_4 = 0$$ We will use these equations to check the word: $011000011= m_1 m_2 m_3 m_4 m_5 r_1 r_2 r_3 r_4$, $$0 + 1 +0+0 + 0 \not \equiv 0$$ $$0 + 1 +0+0 + 0 \not \equiv 0$$ $$0 + 1 +1+0 + 1 \not \equiv 0$$ $$1 + 1 +0+0 + 1 \not \equiv 0$$

Here I ran into a problem because I do not gain any information from these equations as I cannot pinpoint the error. I've checked it several times but can't seem to find the error in my reasoning or how to proceed.

Answer 1

For such a simple code, the simple way to proceed is the following:

You have 32 code words. You can calculate them and check the weight of each word: if each weight is at least three, the code being linear, this implies that each pair of two different code words has a Hamming distance of at least three.

Then, for correcting the word $y = 011000011$: simply calculate its Hamming distance from each code word. You should find one at distance zero or one.

To correct from the syndrome ($H\,y$): by considering all the error positions, you can establish a correspondence (a table) between error positions and the syndrome.