Consider the encoding function $\alpha : \mathbb{Z_2^2} \rightarrow \mathbb{Z_2^5} $ given by the Generator matrix $$ G = \begin{bmatrix}1&0 &1& 0& 0 \\0& 1 & 0 & 1 & 1\end{bmatrix}$$
Now, I need to check that how many error/s can this code detect.
The parity check matrix formed from this is $$H = \begin{bmatrix}1&0 &1& 0& 0 \\0& 1 & 0 & 1 & 0 \\ 0 &1&0&0&1 \end{bmatrix}$$
Now if $H$ has no zero column and no two same columns then I know that atmost two errors can be detected and one error can be corrected. But here, the first and the third column are equal and I don't know what to do in such conditions.
The possible elements of $\mathbb{Z_2^2}$ can be given as $\{ 00, 10, 01, 11\}$, call it $w$ . Now I define $\alpha $ as $\alpha = w.G$ Hence, the possible elements of $\mathbb{Z_2^5}$ are $\{ 00000 , 10100, 01011, 11111 \}$. Ok, so here distances between various codewords can be calculated. But how am I supposed to know that how many errors it can detect?
PS - Although I have accepted an answer but feel free to leave any relevant comment or answer.
The columns of $H$ are all non-zero, so the code detects one error.
But since, as you note, the first and the third column of $H$ are equal, the code does not even correct one error. Just consider the message $00$, which is encoded as $00000$. If there is a transmission error in the first bit, you get $x = 10000$. If there is a transmission error in the third bit you get $y = 00100$.
Now if you multiply $x$ and $y$ by the transpose of $H$, you get the transpose of $100$ in both cases, so you know there is an error, but you are unable to decide whether it occurred in the first or in the third bit.
Seeing this in a different way, you have that $z = 00000$ and $w = 10100$ are both codewords, and you have $$ 10000 = z + 10000 = w + 00100. $$ So if you receive $10000$, you are unable to decide whether the message $z$ has been transmitted, and there was an error in the first bit, or the message $w$ was transmitted, and there was an error in the third bit.
Similarly $$ 00100 = z + 00100 = w + 10000. $$