What are linear codes having minimum distance 2 used for?

Question

Consider the following parity check matrix

$$H = \begin{bmatrix} 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 \end{bmatrix}$$

Since its 1st and 4th column both are identical, the minimum distance of a code is 2. If we compute the syndrome of any received vector $\vec{r}$ and it comes out to be zero, we can't possibly tell whether the received vector is correct or not (as 1st and 4th bit of $\vec{r}$ could be incorrect).

If we can't verify whether the received vector is correct or not, what are linear codes having minimum distance used for?

Answer 1

If the minimum distance is $d=2t+1$ with $t\geq 1$, then the code can correct $t$ errors. The minimum value for $d$ is 3. A code with minimum distance 2 can only detect one error but not correct it.

Answer 2

The typical (practically, the only) linear code with minimum distance 2 is the single parity code which consist of adding a single parity bit, hence $n=k+1$. It detects a single error. See some uses in the page above.

Given that your code is $(6,3)$, it looks rather bad, because for $k=3$ you could have the single parity $(4,3)$ with better rate and same minimum distance. It's true that minimum distance is not the only parameter to judge the quality of the code (more important is the probability of erroneous decoding), but it looks bad anyway.