finding the error pattern from the syndrome

Question

If the parity check matrix is $$ \begin{matrix} 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1\\ \end{matrix} $$ As it has two identical columns how do we decide the error pattern when we get 001 as the syndrome
Is it 01000 or 00010 ?

Answer 1

We can't. To correct one error, the minimum distance has to be $3$. But in a code whose parity check matrix has identical columns, there are codewords of weight $2$, and the minimum distance is therefore at most $2$.

In general, if the parity check matrix of a binary linear code has two identical columns, then the word with $1$s in the positions of the identical columns and $0$s elsewhere is a codeword of weight $2$. In this particular code, $01001$ is such a codeword. But $00000$ is also a codeword, as it is of any linear code. Therefore if the $1$ in the second position of $01001$ is changed to a $0$, the resulting word, $00001$ will be equidistant from the codewords $01001$ and $00000$. One cannot then say what the correct decoding should be.